diff --git a/compiler/testData/codegen/box/delegation/byMiddleInterface.kt b/compiler/testData/codegen/box/delegation/byMiddleInterface.kt
index 5f6256df741..f4e61e75afa 100644
--- a/compiler/testData/codegen/box/delegation/byMiddleInterface.kt
+++ b/compiler/testData/codegen/box/delegation/byMiddleInterface.kt
@@ -1,6 +1,4 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
-
 // FILE: Base.java
 
 public interface Base {
@@ -12,6 +10,7 @@ public interface Base {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
 
 public interface BaseKotlin : Base {
 }
diff --git a/compiler/testData/codegen/box/delegation/defaultOverride.kt b/compiler/testData/codegen/box/delegation/defaultOverride.kt
index c2a66177a7a..25bdbf7a7f2 100644
--- a/compiler/testData/codegen/box/delegation/defaultOverride.kt
+++ b/compiler/testData/codegen/box/delegation/defaultOverride.kt
@@ -1,6 +1,4 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
-
 // FILE: Base.java
 
 public interface Base {
@@ -12,6 +10,7 @@ public interface Base {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
 
 public interface BaseKotlin : Base {
     override fun getValue() = "OK"
diff --git a/compiler/testData/codegen/box/delegation/diamond.kt b/compiler/testData/codegen/box/delegation/diamond.kt
index 8a60bda7ef8..7a539ee55bf 100644
--- a/compiler/testData/codegen/box/delegation/diamond.kt
+++ b/compiler/testData/codegen/box/delegation/diamond.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Base.java
 
@@ -18,8 +17,8 @@ public interface Base2 extends Base {
     }
 }
 
-
 // FILE: main.kt
+// JVM_TARGET: 1.8
 
 interface KBase : Base
 
diff --git a/compiler/testData/codegen/box/delegation/diamond2.kt b/compiler/testData/codegen/box/delegation/diamond2.kt
index daf19245ade..251670895cc 100644
--- a/compiler/testData/codegen/box/delegation/diamond2.kt
+++ b/compiler/testData/codegen/box/delegation/diamond2.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Base.java
 
@@ -18,6 +17,7 @@ public interface Base2 extends Base {
 
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
 
 interface KBase : Base {
     override fun test() = "O" + getValue()
diff --git a/compiler/testData/codegen/box/delegation/inClassDeclaration.kt b/compiler/testData/codegen/box/delegation/inClassDeclaration.kt
index 57d180b8c38..e29eea6a650 100644
--- a/compiler/testData/codegen/box/delegation/inClassDeclaration.kt
+++ b/compiler/testData/codegen/box/delegation/inClassDeclaration.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Base.java
 
@@ -11,6 +10,8 @@ public interface Base {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
+
 class Fail : Base {
     override fun getValue() = "Fail"
 }
diff --git a/compiler/testData/codegen/box/delegation/mixed.kt b/compiler/testData/codegen/box/delegation/mixed.kt
index c8ef7c370a8..091fdbfd6e3 100644
--- a/compiler/testData/codegen/box/delegation/mixed.kt
+++ b/compiler/testData/codegen/box/delegation/mixed.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Base.java
 
@@ -11,6 +10,8 @@ public interface Base extends KBase {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
+
 interface KBase {
     fun getValue(): String
 
diff --git a/compiler/testData/codegen/box/delegation/simple.kt b/compiler/testData/codegen/box/delegation/simple.kt
index 418cea053e1..2f176b6a473 100644
--- a/compiler/testData/codegen/box/delegation/simple.kt
+++ b/compiler/testData/codegen/box/delegation/simple.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Base.java
 
@@ -11,6 +10,8 @@ public interface Base {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
+
 class Fail : Base {
     override fun getValue() = "Fail"
 }
diff --git a/compiler/testData/codegen/box/delegation/simple1.0.kt b/compiler/testData/codegen/box/delegation/simple1.0.kt
index 7395fca1c61..7d22a09f9ed 100644
--- a/compiler/testData/codegen/box/delegation/simple1.0.kt
+++ b/compiler/testData/codegen/box/delegation/simple1.0.kt
@@ -1,6 +1,5 @@
 // !LANGUAGE: -NoDelegationToJavaDefaultInterfaceMembers
 // IGNORE_BACKEND_FIR: JVM_IR
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Base.java
 
@@ -13,6 +12,8 @@ public interface Base {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
+
 class OK : Base {
     override fun getValue() = "OK"
 }
diff --git a/compiler/testData/codegen/box/javaInterop/notNullAssertions/enhancedNullability/inFunctionWithExpressionBodyWithJavaGeneric.kt b/compiler/testData/codegen/box/javaInterop/notNullAssertions/enhancedNullability/inFunctionWithExpressionBodyWithJavaGeneric.kt
index e85957c8e19..b979afffc2e 100644
--- a/compiler/testData/codegen/box/javaInterop/notNullAssertions/enhancedNullability/inFunctionWithExpressionBodyWithJavaGeneric.kt
+++ b/compiler/testData/codegen/box/javaInterop/notNullAssertions/enhancedNullability/inFunctionWithExpressionBodyWithJavaGeneric.kt
@@ -1,6 +1,5 @@
 // !LANGUAGE: +StrictJavaNullabilityAssertions
 // TARGET_BACKEND: JVM
-// SKIP_JDK6
 // See KT-8135
 // We could generate runtime assertion on call site for 'generic<NOT_NULL_TYPE>()' below.
 
@@ -24,11 +23,12 @@ fun <T> generic(j: J) = j.nullT<T>()
 import org.jetbrains.annotations.NotNull;
 
 public class J {
-    public <T> @NotNull T nullT() {
+    @NotNull
+    public <T> T nullT() {
         return null;
     }
 
     public void test() {
         TestKt.withAssertion(this);
     }
-}
\ No newline at end of file
+}
diff --git a/compiler/testData/codegen/box/jvm8/javaDefaults/defaultMethodOverride.kt b/compiler/testData/codegen/box/jvm8/javaDefaults/defaultMethodOverride.kt
index 287f7458c32..6deb9b3d6e9 100644
--- a/compiler/testData/codegen/box/jvm8/javaDefaults/defaultMethodOverride.kt
+++ b/compiler/testData/codegen/box/jvm8/javaDefaults/defaultMethodOverride.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Simple.java
 
@@ -9,6 +8,7 @@ public interface Simple {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
 
 interface KInterface: Simple {
     override fun test(s: String): String {
diff --git a/compiler/testData/codegen/box/jvm8/javaDefaults/dontDelegateToDefaultMethods.kt b/compiler/testData/codegen/box/jvm8/javaDefaults/dontDelegateToDefaultMethods.kt
index 148aaf82fed..b1d313c5b4b 100644
--- a/compiler/testData/codegen/box/jvm8/javaDefaults/dontDelegateToDefaultMethods.kt
+++ b/compiler/testData/codegen/box/jvm8/javaDefaults/dontDelegateToDefaultMethods.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: Test.java
 
@@ -12,6 +11,7 @@ interface Test<T> {
 }
 
 // FILE: main.kt
+// JVM_TARGET: 1.8
 
 class Child : Test<String> {
     override fun call() : String {
diff --git a/compiler/testData/codegen/box/jvm8/javaDefaults/samOnInterfaceWithDefaultMethod.kt b/compiler/testData/codegen/box/jvm8/javaDefaults/samOnInterfaceWithDefaultMethod.kt
index a69fb2b71df..9afd98732b1 100644
--- a/compiler/testData/codegen/box/jvm8/javaDefaults/samOnInterfaceWithDefaultMethod.kt
+++ b/compiler/testData/codegen/box/jvm8/javaDefaults/samOnInterfaceWithDefaultMethod.kt
@@ -1,4 +1,3 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
 // FILE: JavaCall.java
 
@@ -24,6 +23,7 @@ interface Test {
 }
 
 // FILE: sam.kt
+// JVM_TARGET: 1.8
 
 fun box(): String {
     val lambda = { "X" }
diff --git a/compiler/testData/codegen/box/smartCasts/kt19058.kt b/compiler/testData/codegen/box/smartCasts/kt19058.kt
index 1f558682449..8f92195d0f2 100644
--- a/compiler/testData/codegen/box/smartCasts/kt19058.kt
+++ b/compiler/testData/codegen/box/smartCasts/kt19058.kt
@@ -1,5 +1,5 @@
-// SKIP_JDK6
 // TARGET_BACKEND: JVM
+// JVM_TARGET: 1.8
 // FILE: Test.kt
 open class KFoo {
     fun foo(): String {
@@ -17,4 +17,4 @@ public interface JBar {
     default String getBar() {
         return "OK";
     }
-}
\ No newline at end of file
+}
diff --git a/compiler/testData/codegen/box/traits/multipleImplFromJava.kt b/compiler/testData/codegen/box/traits/multipleImplFromJava.kt
index 748dceeecb8..276f901456f 100644
--- a/compiler/testData/codegen/box/traits/multipleImplFromJava.kt
+++ b/compiler/testData/codegen/box/traits/multipleImplFromJava.kt
@@ -1,5 +1,4 @@
 // TARGET_BACKEND: JVM
-// SKIP_JDK6
 // FILE: I.java
 interface I {
     default String ifun() { return "fail"; }
@@ -14,6 +13,7 @@ public class Z {
 public class Zz extends Z implements I {}
 
 // FILE: multipleImplFromJava.kt
+// JVM_TARGET: 1.8
 
 class Cc : Zz()
 
diff --git a/compiler/testData/codegen/bytecodeListing/kt43217.kt b/compiler/testData/codegen/bytecodeListing/kt43217.kt
index a62b2547b78..966f20378c3 100644
--- a/compiler/testData/codegen/bytecodeListing/kt43217.kt
+++ b/compiler/testData/codegen/bytecodeListing/kt43217.kt
@@ -1,4 +1,5 @@
 // FILE: kt43217.kt
+// JVM_TARGET: 1.8
 class A {
     private val b =
         object : DoubleExpression() {
@@ -43,4 +44,3 @@ public interface ObservableDouble extends ObservableValue<Double> {
     @NotNull
     Object isEqualTo(double value);
 }
-
diff --git a/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/AbstractBlackBoxAgainstJavaCodegenTest.kt b/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/AbstractBlackBoxAgainstJavaCodegenTest.kt
index b45162d67e1..e7805127728 100644
--- a/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/AbstractBlackBoxAgainstJavaCodegenTest.kt
+++ b/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/AbstractBlackBoxAgainstJavaCodegenTest.kt
@@ -18,12 +18,17 @@ package org.jetbrains.kotlin.codegen
 
 import org.jetbrains.kotlin.cli.jvm.config.addJvmClasspathRoot
 import org.jetbrains.kotlin.config.CompilerConfiguration
+import org.jetbrains.kotlin.config.JvmTarget
 import java.io.File
 
 abstract class AbstractBlackBoxAgainstJavaCodegenTest : AbstractBlackBoxCodegenTest() {
     override fun doMultiFileTest(wholeFile: File, files: List<TestFile>) {
         javaClassesOutputDirectory = writeJavaFiles(files)!!.let { directory ->
-            CodegenTestUtil.compileJava(CodegenTestUtil.findJavaSourcesInDirectory(directory), emptyList(), extractJavacOptions(files))
+            val jvmTargets = files.mapNotNullTo(mutableSetOf()) { it.directives["JVM_TARGET"]?.let((JvmTarget)::fromString) }
+            check(jvmTargets.size <= 1) { "Having different JVM_TARGETs for different files is not supported in this test." }
+            CodegenTestUtil.compileJava(
+                CodegenTestUtil.findJavaSourcesInDirectory(directory), emptyList(), extractJavacOptions(files, jvmTargets.firstOrNull())
+            )
         }
 
         super.doMultiFileTest(wholeFile, files.map { file ->
diff --git a/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/CodegenTestCase.java b/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/CodegenTestCase.java
index 5e1fddd142e..9debd333195 100644
--- a/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/CodegenTestCase.java
+++ b/compiler/tests-common/tests/org/jetbrains/kotlin/codegen/CodegenTestCase.java
@@ -63,9 +63,7 @@ import static org.jetbrains.kotlin.cli.common.output.OutputUtilsKt.writeAllTo;
 import static org.jetbrains.kotlin.codegen.CodegenTestUtil.*;
 import static org.jetbrains.kotlin.codegen.TestUtilsKt.extractUrls;
 import static org.jetbrains.kotlin.test.KotlinTestUtils.getAnnotationsJar;
-import static org.jetbrains.kotlin.test.clientserver.TestProcessServerKt.getBoxMethodOrNull;
-import static org.jetbrains.kotlin.test.clientserver.TestProcessServerKt.getGeneratedClass;
-import static org.jetbrains.kotlin.test.clientserver.TestProcessServerKt.runBoxMethod;
+import static org.jetbrains.kotlin.test.clientserver.TestProcessServerKt.*;
 
 public abstract class CodegenTestCase extends KotlinBaseTest<KotlinBaseTest.TestFile> {
     private static final String DEFAULT_TEST_FILE_NAME = "a_test";
@@ -487,7 +485,6 @@ public abstract class CodegenTestCase extends KotlinBaseTest<KotlinBaseTest.Test
                 it -> InTextDirectivesUtils.isDirectiveDefined(it.content, "ANDROID_ANNOTATIONS")
         );
 
-        List<String> javacOptions = extractJavacOptions(files);
         List<File> classpath = new ArrayList<>();
         classpath.add(getAnnotationsJar());
 
@@ -531,6 +528,7 @@ public abstract class CodegenTestCase extends KotlinBaseTest<KotlinBaseTest.Test
             updateJavaClasspath(javaClasspath);
 
             javaClassesOutputDirectory = getJavaClassesOutputDirectory();
+            List<String> javacOptions = extractJavacOptions(files, configuration.get(JVMConfigurationKeys.JVM_TARGET));
             compileJava(findJavaSourcesInDirectory(javaSourceDir), javaClasspath, javacOptions, javaClassesOutputDirectory);
         }
     }
@@ -538,22 +536,33 @@ public abstract class CodegenTestCase extends KotlinBaseTest<KotlinBaseTest.Test
     protected void updateJavaClasspath(@NotNull List<String> javaClasspath) {}
 
     @NotNull
-    protected static List<String> extractJavacOptions(@NotNull List<TestFile> files) {
+    protected static List<String> extractJavacOptions(@NotNull List<TestFile> files, @Nullable JvmTarget kotlinTarget) {
         List<String> javacOptions = new ArrayList<>(0);
         for (TestFile file : files) {
             javacOptions.addAll(InTextDirectivesUtils.findListWithPrefixes(file.content, "// JAVAC_OPTIONS:"));
         }
-        updateJavacOptions(javacOptions);
+        String javaTarget = computeJavaTarget(javacOptions, kotlinTarget);
+        if (javaTarget != null) {
+            javacOptions.add("-source");
+            javacOptions.add(javaTarget);
+            javacOptions.add("-target");
+            javacOptions.add(javaTarget);
+        }
         return javacOptions;
     }
 
-    private static void updateJavacOptions(@NotNull List<String> javacOptions) {
-        if (JAVA_COMPILATION_TARGET != null && !javacOptions.contains("-target")) {
-            javacOptions.add("-source");
-            javacOptions.add(JAVA_COMPILATION_TARGET);
-            javacOptions.add("-target");
-            javacOptions.add(JAVA_COMPILATION_TARGET);
-        }
+    private static final boolean IS_SOURCE_6_STILL_SUPPORTED =
+            // JDKs up to 11 do support -source/target 1.6, but later -- don't.
+            Arrays.asList("1.6", "1.7", "1.8", "9", "10", "11").contains(System.getProperty("java.specification.version"));
+
+    private static String computeJavaTarget(@NotNull List<String> javacOptions, @Nullable JvmTarget kotlinTarget) {
+        if (JAVA_COMPILATION_TARGET != null && !javacOptions.contains("-target"))
+            return JAVA_COMPILATION_TARGET;
+        if (kotlinTarget != null && kotlinTarget.compareTo(JvmTarget.JVM_1_6) > 0)
+            return kotlinTarget.getDescription();
+        if (IS_SOURCE_6_STILL_SUPPORTED)
+            return "1.6";
+        return null;
     }
 
     @NotNull