diff --git a/core/modules/system/tests/fixtures/update/drupal-8.update-test-semver-update-n-enabled.php b/core/modules/system/tests/fixtures/update/drupal-8.update-test-semver-update-n-enabled.php
new file mode 100644
index 0000000000..663360955a
--- /dev/null
+++ b/core/modules/system/tests/fixtures/update/drupal-8.update-test-semver-update-n-enabled.php
@@ -0,0 +1,38 @@
+<?php
+
+/**
+ * @file
+ * Partial database to mimic the installation of the update_test_schema module.
+ */
+
+use Drupal\Core\Database\Database;
+
+$connection = Database::getConnection();
+
+// Set the schema version.
+$connection->merge('key_value')
+  ->condition('collection', 'system.schema')
+  ->condition('name', 'update_test_semver_update_n')
+  ->fields([
+    'collection' => 'system.schema',
+    'name' => 'update_test_semver_update_n',
+    'value' => 'i:8000;',
+  ])
+  ->execute();
+
+// Update core.extension.
+$extensions = $connection->select('config')
+  ->fields('config', ['data'])
+  ->condition('collection', '')
+  ->condition('name', 'core.extension')
+  ->execute()
+  ->fetchField();
+$extensions = unserialize($extensions);
+$extensions['module']['update_test_semver_update_n'] = 8000;
+$connection->update('config')
+  ->fields([
+    'data' => serialize($extensions),
+  ])
+  ->condition('collection', '')
+  ->condition('name', 'core.extension')
+  ->execute();
diff --git a/core/modules/system/tests/modules/system_core_semver_test/system_core_semver_test.info.yml b/core/modules/system/tests/modules/system_core_semver_test/system_core_semver_test.info.yml
new file mode 100644
index 0000000000..8063e099b7
--- /dev/null
+++ b/core/modules/system/tests/modules/system_core_semver_test/system_core_semver_test.info.yml
@@ -0,0 +1,6 @@
+name: 'System core ^8 version test'
+type: module
+description: 'Support module for testing core using semver.'
+package: Testing
+version: 1.0.0
+core: ^8
diff --git a/core/modules/system/tests/modules/system_incompatible_core_version_test_9x/system_incompatible_core_version_test_9x.info.yml b/core/modules/system/tests/modules/system_incompatible_core_version_test_9x/system_incompatible_core_version_test_99x.info.yml
similarity index 71%
rename from core/modules/system/tests/modules/system_incompatible_core_version_test_9x/system_incompatible_core_version_test_9x.info.yml
rename to core/modules/system/tests/modules/system_incompatible_core_version_test_9x/system_incompatible_core_version_test_99x.info.yml
index 2f9f9d5750..449c61b3ce 100644
--- a/core/modules/system/tests/modules/system_incompatible_core_version_test_9x/system_incompatible_core_version_test_9x.info.yml
+++ b/core/modules/system/tests/modules/system_incompatible_core_version_test_9x/system_incompatible_core_version_test_99x.info.yml
@@ -1,4 +1,4 @@
-name: 'System incompatible core 9.x version test'
+name: 'System incompatible core 99.x version test'
 type: module
 description: 'Support module for testing system core incompatibility.'
 package: Testing
diff --git a/core/modules/system/tests/modules/system_test/system_test.module b/core/modules/system/tests/modules/system_test/system_test.module
index e0825d2993..e90b54a85b 100644
--- a/core/modules/system/tests/modules/system_test/system_test.module
+++ b/core/modules/system/tests/modules/system_test/system_test.module
@@ -75,7 +75,7 @@ function system_test_system_info_alter(&$info, Extension $file, $type) {
     'system_incompatible_core_version_dependencies_test',
     'system_incompatible_module_version_test',
     'system_incompatible_core_version_test',
-    'system_incompatible_core_version_test_9x',
+    'system_incompatible_core_version_test_99x',
   ])) {
     $info['hidden'] = FALSE;
   }
diff --git a/core/modules/system/tests/modules/update_test_semver_update_n/update_test_semver_update_n.info.yml b/core/modules/system/tests/modules/update_test_semver_update_n/update_test_semver_update_n.info.yml
new file mode 100644
index 0000000000..5465a62d0a
--- /dev/null
+++ b/core/modules/system/tests/modules/update_test_semver_update_n/update_test_semver_update_n.info.yml
@@ -0,0 +1,6 @@
+name: 'Update test hook_update_n semver'
+type: module
+description: 'Support module for update testing with core semver value.'
+package: Testing
+version: VERSION
+core: ^8
diff --git a/core/modules/system/tests/modules/update_test_semver_update_n/update_test_semver_update_n.install b/core/modules/system/tests/modules/update_test_semver_update_n/update_test_semver_update_n.install
new file mode 100644
index 0000000000..2f240eeb01
--- /dev/null
+++ b/core/modules/system/tests/modules/update_test_semver_update_n/update_test_semver_update_n.install
@@ -0,0 +1,17 @@
+<?php
+
+/**
+ * @file
+ * Update hooks for the update_test_semver_update_n module.
+ */
+
+use Drupal\Core\Database\Database;
+use Drupal\Core\Url;
+use Drupal\Component\Render\FormattableMarkup;
+
+/**
+ * Update 8002.
+ */
+function update_test_semver_update_n_update_8001() {
+  \Drupal::state()->set('update_test_semver_update_n_update_8001', 'Yes, I was run. Thanks for testing!');
+}
diff --git a/core/modules/system/tests/src/Functional/Module/DependencyTest.php b/core/modules/system/tests/src/Functional/Module/DependencyTest.php
index d5b0a78ee8..58979081c0 100644
--- a/core/modules/system/tests/src/Functional/Module/DependencyTest.php
+++ b/core/modules/system/tests/src/Functional/Module/DependencyTest.php
@@ -115,7 +115,7 @@ public function testCoreVersionDependency() {
     // Test the next minor release.
     \Drupal::state()->set('dependency_test.core_version_requirement', "~$major.$next_minor");
     $this->drupalGet('admin/modules');
-    $assert_session->fieldDisabled('modules[system_incompatible_core_version_test_9x][enable]');
+    $assert_session->fieldDisabled('modules[system_incompatible_core_version_test_99x][enable]');
     $assert_session->fieldDisabled('modules[common_test][enable]');
 
     // Test either current major or the next one.
@@ -137,11 +137,13 @@ public function testCoreVersionDependency() {
     \Drupal::state()->set('dependency_test.core_version_requirement', "~$major.$minor");
     $this->drupalGet('admin/modules');
     $this->assertFalse($assert_session->elementExists('css', '[name="modules[common_test][enable]"]')->hasAttribute('disabled'));
+    $this->assertFalse($assert_session->elementExists('css', '[name="modules[system_core_semver_test][enable]"]')->hasAttribute('disabled'));
 
-    // Ensure the module can actually be installed.
+    // Ensure the modules can actually be installed.
     $edit['modules[common_test][enable]'] = 'common_test';
+    $edit['modules[system_core_semver_test][enable]'] = 'system_core_semver_test';
     $this->drupalPostForm('admin/modules', $edit, t('Install'));
-    $this->assertModules(['common_test'], TRUE);
+    $this->assertModules(['common_test', 'system_core_semver_test'], TRUE);
   }
 
   /**
diff --git a/core/modules/system/tests/src/Functional/System/ThemeTest.php b/core/modules/system/tests/src/Functional/System/ThemeTest.php
index 02952f6656..0c6934624f 100644
--- a/core/modules/system/tests/src/Functional/System/ThemeTest.php
+++ b/core/modules/system/tests/src/Functional/System/ThemeTest.php
@@ -367,8 +367,11 @@ public function testInvalidTheme() {
     $this->assertText(t('This theme requires the base theme @base_theme to operate correctly.', ['@base_theme' => 'not_real_test_basetheme']));
     $this->assertText(t('This theme requires the base theme @base_theme to operate correctly.', ['@base_theme' => 'test_invalid_basetheme']));
     $this->assertText(t('This theme requires the theme engine @theme_engine to operate correctly.', ['@theme_engine' => 'not_real_engine']));
-    // Check for the error text of a theme with the wrong core version.
-    $this->assertText("This theme is not compatible with Drupal " . \Drupal::VERSION . ". Check that the .info.yml file contains a compatible 'core' value.");
+    // Check for the error text of a theme with the wrong core version
+    // using 7.x and ^7.
+    $incompatible_core_message = 'This theme is not compatible with Drupal ' . \Drupal::VERSION . ". Check that the .info.yml file contains a compatible 'core' value.";
+    $this->assertThemeIncompatibleText('Theme test with invalid core version 8.8.0-dev', $incompatible_core_message);
+    $this->assertThemeIncompatibleText('Theme test with invalid semver core version', $incompatible_core_message);
     // Check for the error text of a theme without a content region.
     $this->assertText("This theme is missing a 'content' region.");
   }
@@ -437,24 +440,29 @@ public function testUninstallingThemes() {
    * Tests installing a theme and setting it as default.
    */
   public function testInstallAndSetAsDefault() {
-    $this->drupalGet('admin/appearance');
-    // Bartik is uninstalled in the test profile and has the third "Install and
-    // set as default" link.
-    $this->clickLink(t('Install and set as default'), 2);
-    // Test the confirmation message.
-    $this->assertText('Bartik is now the default theme.');
-    // Make sure Bartik is now set as the default theme in config.
-    $this->assertEqual($this->config('system.theme')->get('default'), 'bartik');
-
-    // This checks for a regression. See https://www.drupal.org/node/2498691.
-    $this->assertNoText('The bartik theme was not found.');
-
-    $themes = \Drupal::service('theme_handler')->rebuildThemeData();
-    $version = $themes['bartik']->info['version'];
+    $themes = [
+      'bartik' => 'Bartik',
+      'test_core_semver' => 'Theme test with semver core version',
+    ];
+    foreach ($themes as $theme_machine_name => $theme_name) {
+      $this->drupalGet('admin/appearance');
+      $this->getSession()->getPage()->findLink("Install $theme_name as default theme")->click();
+      // Test the confirmation message.
+      $this->assertText("$theme_name is now the default theme.");
+      // Make sure Bartik is now set as the default theme in config.
+      $this->assertEqual($this->config('system.theme')->get('default'), $theme_machine_name);
+
+      // This checks for a regression. See https://www.drupal.org/node/2498691.
+      $this->assertNoText("The $theme_machine_name theme was not found.");
+
+      $themes = \Drupal::service('theme_handler')->rebuildThemeData();
+      $version = $themes[$theme_machine_name]->info['version'];
+
+      // Confirm Bartik is indicated as the default theme.
+      $out = $this->getSession()->getPage()->getContent();
+      $this->assertTrue((bool) preg_match("/$theme_name " . preg_quote($version) . '\s{2,}\(default theme\)/', $out));
+    }
 
-    // Confirm Bartik is indicated as the default theme.
-    $out = $this->getSession()->getPage()->getContent();
-    $this->assertTrue((bool) preg_match('/Bartik ' . preg_quote($version) . '\s{2,}\(default theme\)/', $out));
   }
 
   /**
@@ -470,4 +478,8 @@ public function testThemeSettingsNoLogoNoFavicon() {
     $this->assertText('The configuration options have been saved.');
   }
 
+  private function assertThemeIncompatibleText($theme_name, $expected_text) {
+    $this->assertSession()->elementExists('css', ".theme-info:contains(\"$theme_name\") .incompatible:contains(\"$expected_text\")");
+  }
+
 }
diff --git a/core/modules/system/tests/themes/test_core_semver/test_core_semver.info.yml b/core/modules/system/tests/themes/test_core_semver/test_core_semver.info.yml
new file mode 100644
index 0000000000..2ba1eb76e9
--- /dev/null
+++ b/core/modules/system/tests/themes/test_core_semver/test_core_semver.info.yml
@@ -0,0 +1,5 @@
+name: 'Theme test with semver core version'
+type: theme
+description: 'Test theme which has semver core version.'
+version: VERSION
+core: ^8 || ^9
diff --git a/core/modules/system/tests/themes/test_invalid_core_semver/test_invalid_core_semver.info.yml b/core/modules/system/tests/themes/test_invalid_core_semver/test_invalid_core_semver.info.yml
new file mode 100644
index 0000000000..f349cfc0f9
--- /dev/null
+++ b/core/modules/system/tests/themes/test_invalid_core_semver/test_invalid_core_semver.info.yml
@@ -0,0 +1,5 @@
+name: 'Theme test with invalid semver core version'
+type: theme
+description: 'Test theme which has an invalid semver core version.'
+version: VERSION
+core: ^7
diff --git a/core/tests/Drupal/FunctionalTests/Update/UpdatePathTestBaseTest.php b/core/tests/Drupal/FunctionalTests/Update/UpdatePathTestBaseTest.php
index 942cb886ad..504eb96ab9 100644
--- a/core/tests/Drupal/FunctionalTests/Update/UpdatePathTestBaseTest.php
+++ b/core/tests/Drupal/FunctionalTests/Update/UpdatePathTestBaseTest.php
@@ -26,6 +26,7 @@ protected function setDatabaseDumpFiles() {
     $this->databaseDumpFiles = [
       __DIR__ . '/../../../../modules/system/tests/fixtures/update/drupal-8.bare.standard.php.gz',
       __DIR__ . '/../../../../modules/system/tests/fixtures/update/drupal-8.update-test-schema-enabled.php',
+      __DIR__ . '/../../../../modules/system/tests/fixtures/update/drupal-8.update-test-semver-update-n-enabled.php',
     ];
   }
 
@@ -99,8 +100,11 @@ public function testUpdateHookN() {
 
     // Ensure schema has changed.
     $this->assertEqual(drupal_get_installed_schema_version('update_test_schema', TRUE), 8001);
+    $this->assertEqual(drupal_get_installed_schema_version('update_test_semver_update_n', TRUE), 8001);
     // Ensure the index was added for column a.
     $this->assertTrue($connection->schema()->indexExists('update_test_schema_table', 'test'), 'Version 8001 of the update_test_schema module is installed.');
+    // Ensure update_test_semver_update_n_update_8001 was run.
+    $this->assertEquals(\Drupal::state()->get('update_test_semver_update_n_update_8001'), 'Yes, I was run. Thanks for testing!');
   }
 
   /**