diff --git a/.github/workflows/Test.yml b/.github/workflows/Test.yml
index 7a773bf..6cb3a77 100644
--- a/.github/workflows/Test.yml
+++ b/.github/workflows/Test.yml
@@ -18,10 +18,13 @@ jobs:
         strategy:
             fail-fast: false
             matrix:
-                typo3: [ '^11.5.0', '^12.4.0' ]
+                typo3: [ '^11.5.0', '^12.4.0', '^13.4.0' ]
                 php: [ '8.1', '8.2', '8.3' ]
                 dependency-version: [ lowest, stable ]
                 experimental: [ false ]
+                exclude:
+                    - typo3: '^13.4.0'
+                      php: '8.1'
 
         continue-on-error: ${{ matrix.experimental }}
 
diff --git a/composer.json b/composer.json
index 8acdc0b..ae1b5c2 100644
--- a/composer.json
+++ b/composer.json
@@ -16,7 +16,7 @@
         "symfony/console": "^5.4 || ^6.4 || ^7.0",
         "symfony/yaml": "^5.4 || ^6.4 || ^7.0",
         "typo3/cms-composer-installers": "^4.0@rc || >=5.0",
-        "typo3/cms-core": "^11.5.20 || ^12.4",
+        "typo3/cms-core": "^11.5.20 || ^12.4 || ^13.4",
         "composer-runtime-api": "^2.2",
         "symfony/polyfill-php80": "^1.23.1"
     },
diff --git a/src/Composer/InstallerScript/DumpSettings.php b/src/Composer/InstallerScript/DumpSettings.php
index ee95177..a82d77a 100644
--- a/src/Composer/InstallerScript/DumpSettings.php
+++ b/src/Composer/InstallerScript/DumpSettings.php
@@ -45,7 +45,7 @@ private function generateLocalConfigurationFile(ScriptEvent $event): bool
         $fileSystem = new Filesystem();
         $fileSystem->ensureDirectoryExists(getenv('TYPO3_PATH_APP') . '/config');
         $fileSystem->ensureDirectoryExists(getenv('TYPO3_PATH_APP') . '/var/cache/code');
-        if ($this->isMajorVersionTwelve($event->getComposer())) {
+        if ($this->isMajorVersionTwelveOrHigher($event->getComposer())) {
             $systemConfigurationFile = getenv('TYPO3_PATH_APP') . '/config/system/settings.php';
             $fileSystem->ensureDirectoryExists(getenv('TYPO3_PATH_APP') . '/config/system');
         } else {
@@ -100,7 +100,7 @@ private function generateLocalConfigurationFile(ScriptEvent $event): bool
         ) > 0;
     }
 
-    private function isMajorVersionTwelve(Composer $composer): bool
+    private function isMajorVersionTwelveOrHigher(Composer $composer): bool
     {
         $repository = $composer->getRepositoryManager()->getLocalRepository();
         $corePackage = $repository->findPackage('typo3/cms-core', '*');
diff --git a/src/Typo3SiteConfiguration.php b/src/Typo3SiteConfiguration.php
index 08b9b93..27ec9d1 100644
--- a/src/Typo3SiteConfiguration.php
+++ b/src/Typo3SiteConfiguration.php
@@ -12,6 +12,9 @@
 
 class Typo3SiteConfiguration extends SiteConfiguration
 {
+    protected $configFileName = 'config.yaml';
+    protected $cacheIdentifier = 'sites-configuration';
+
     /**
      * Load plain configuration
      * This method should only be used in case the original configuration as it exists in the file should be loaded,
diff --git a/src/Xclass/ConfigurationManager.php b/src/Xclass/ConfigurationManager.php
index 68ce9d3..3f8be96 100644
--- a/src/Xclass/ConfigurationManager.php
+++ b/src/Xclass/ConfigurationManager.php
@@ -55,14 +55,13 @@
  *
  * This class handles the access to the files
  * - EXT:core/Configuration/DefaultConfiguration.php (default TYPO3_CONF_VARS)
- * - typo3conf/LocalConfiguration.php (overrides of TYPO3_CONF_VARS)
- * - typo3conf/AdditionalConfiguration.php (optional additional local code blocks)
+ * - config/system/settings.php or typo3conf/system/settings.php - previously known as LocalConfiguration.php
+ * - config/system/additional.php or typo3conf/system/additional.php (optional additional code blocks) - previously known as typo3conf/AdditionalConfiguration.php
  *
  * IMPORTANT:
  *   This class is intended for internal core use ONLY.
  *   Extensions should usually use the resulting $GLOBALS['TYPO3_CONF_VARS'] array,
- *   do not try to modify settings in LocalConfiguration.php with an extension.
- *
+ *   do not try to modify settings in the config/system/settings.php file with an extension.
  * @internal
  */
 class ConfigurationManager
@@ -130,7 +129,7 @@ class ConfigurationManager
      *
      * @var array
      */
-    protected $whiteListedLocalConfigurationPaths = [
+    protected array $allowedSettingsPaths = [
         'EXTCONF',
         'DB',
         'SYS/caching/cacheConfigurations',
@@ -190,7 +189,8 @@ public function getDefaultConfigurationDescriptionFileLocation()
     }
 
     /**
-     * Return local configuration array typo3conf/LocalConfiguration.php
+     * Return configuration array of typo3conf/system/settings.php or config/system/settings.php, falls back
+     * to typo3conf/LocalConfiguration.php
      *
      * @return array Content array of local configuration file
      */
@@ -466,7 +466,7 @@ public function enableFeature(string $featureName): bool
     }
 
     /**
-     * Disables a feature and writes the option to LocalConfiguration.php
+     * Disables a feature and writes the option to system/settings.php
      * Short-hand method
      * Warning: TO BE USED ONLY to disable a single feature.
      * NOT TO BE USED within iterations to disable multiple features.
@@ -509,7 +509,7 @@ public function exportConfiguration()
     }
 
     /**
-     * Write local configuration array to typo3conf/LocalConfiguration.php
+     * Write configuration array to %config-dir%/system/settings.php
      *
      * @param array $configuration The local configuration to be written
      *
@@ -529,7 +529,7 @@ public function writeLocalConfiguration(array $configuration)
     }
 
     /**
-     * Write additional configuration array to typo3conf/AdditionalConfiguration.php
+     * Write additional configuration array to config/system/additional.php / typo3conf/system/additional.php
      *
      * @param array $additionalConfigurationLines The configuration lines to be written
      *
@@ -549,18 +549,17 @@ public function writeAdditionalConfiguration(array $additionalConfigurationLines
 
     /**
      * Uses FactoryConfiguration file and a possible AdditionalFactoryConfiguration
-     * file in typo3conf to create a basic LocalConfiguration.php. This is used
-     * by the install tool in an early step.
+     * file in typo3conf to create a basic config/system/settings.php. This is used
+     * by the installer in an early step.
      *
      * @throws \RuntimeException
-     *
      * @internal
      */
     public function createLocalConfigurationFromFactoryConfiguration()
     {
         if (file_exists($this->getLocalConfigurationFileLocation())) {
             throw new \RuntimeException(
-                'LocalConfiguration.php exists already',
+                basename($this->getSystemConfigurationFileLocation(true)) . ' already exists',
                 1364836026
             );
         }
@@ -583,14 +582,13 @@ public function createLocalConfigurationFromFactoryConfiguration()
      * Check if access / write to given path in local configuration is allowed.
      *
      * @param string $path Path to search for
-     *
      * @return bool TRUE if access is allowed
      */
     protected function isValidLocalConfigurationPath($path)
     {
         // Early return for white listed paths
-        foreach ($this->whiteListedLocalConfigurationPaths as $whiteListedPath) {
-            if (str_starts_with($path, $whiteListedPath)) {
+        foreach ($this->allowedSettingsPaths as $allowedSettingsPath) {
+            if (str_starts_with($path, $allowedSettingsPath)) {
                 return true;
             }
         }