EntityManager has been split into 11 classes

The EntityManager class was originally written to provide discovery for entity types.
It grew to be responsible for all methods around entities, entity types, entity fields, and entity type bundles.

In order to maintain backwards-compatibility, the EntityManager class remains intact, but contains no concrete method implementations, instead delegating all method calls to the appropriate service.

Following is a list of methods that were previously on EntityManagerInterface, organized by the interface they are now on.

Interface: \Drupal\Core\Entity\EntityTypeManagerInterface
Class: \Drupal\Core\Entity\EntityTypeManager
Service: entity_type.manager
Methods:
  public function getAccessControlHandler($entity_type);
  public function getStorage($entity_type);
  public function getViewBuilder($entity_type);
  public function getListBuilder($entity_type);
  public function getFormObject($entity_type, $operation);
  public function getRouteProviders($entity_type);
  public function hasHandler($entity_type, $handler_type);
  public function getHandler($entity_type, $handler_type);
  public function createHandlerInstance($class, EntityTypeInterface $definition = NULL);
  public function getDefinition($entity_type_id, $exception_on_invalid = TRUE);
  public function getDefinitions();

Interface: \Drupal\Core\Entity\EntityTypeRepositoryInterface
Class: \Drupal\Core\Entity\EntityTypeRepository
Service: entity_type.repository
Methods:
  public function getEntityTypeLabels($group = FALSE);
  public function getEntityTypeFromClass($class_name);

Interface: \Drupal\Core\Entity\EntityTypeBundleInfoInterface
Class: \Drupal\Core\Entity\EntityTypeBundleInfo
Service: entity_type.bundle.info
Methods:
  public function getAllBundleInfo();
  public function getBundleInfo($entity_type);
  public function clearCachedBundles();

Interface: \Drupal\Core\Entity\EntityDisplayRepositoryInterface
Class: \Drupal\Core\Entity\EntityDisplayRepository
Service: entity_display.repository
Methods:
  public function getAllViewModes();
  public function getViewModes($entity_type_id);
  public function getAllFormModes();
  public function getFormModes($entity_type_id);
  public function getViewModeOptions($entity_type_id, $include_disabled = FALSE);
  public function getFormModeOptions($entity_type_id, $include_disabled = FALSE);
  public function getViewModeOptionsByBundle($entity_type_id, $bundle);
  public function getFormModeOptionsByBundle($entity_type_id, $bundle);
  public function clearDisplayModeInfo();

Interface: \Drupal\Core\Entity\EntityFieldManagerInterface
Class: \Drupal\Core\Entity\EntityFieldManager
Service: entity_field.manager
Methods:
  public function getBaseFieldDefinitions($entity_type_id);
  public function getFieldDefinitions($entity_type_id, $bundle);
  public function getFieldStorageDefinitions($entity_type_id);
  public function getFieldMap();
  public function setFieldMap(array $field_map);
  public function getFieldMapByFieldType($field_type);
  public function clearCachedFieldDefinitions();
  public function useCaches($use_caches = FALSE);
  public function getExtraFields($entity_type_id, $bundle);

Interface: \Drupal\Core\Entity\EntityTypeListenerInterface
Class: \Drupal\Core\Entity\EntityTypeListener
Service: entity_type.listener
Methods:
  public function onEntityTypeCreate(EntityTypeInterface $entity_type);
  public function onEntityTypeUpdate(EntityTypeInterface $entity_type, EntityTypeInterface $original);
  public function onEntityTypeDelete(EntityTypeInterface $entity_type);

Interface: \Drupal\Core\Entity\EntityInstalledDefinitionRepositoryInterface
Class: \Drupal\Core\Entity\EntityInstalledDefinitionRepository
Service: entity_definition.repository
Methods:
  public function getLastInstalledDefinition($entity_type_id);
  public function getLastInstalledFieldStorageDefinitions($entity_type_id);

Interface: \Drupal\Core\Entity\EntityRepositoryInterface
Class: \Drupal\Core\Entity\EntityRepository
Service: entity.repository
Methods:
  public function loadEntityByUuid($entity_type_id, $uuid);
  public function loadEntityByConfigTarget($entity_type_id, $target);
  public function getTranslationFromContext(EntityInterface $entity, $langcode = NULL, $context = array());

Interface: \Drupal\Core\Entity\EntityBundleListenerInterface
Class: \Drupal\Core\Entity\EntityBundleListener
Service: entity_bundle.listener
Methods:
  public function onBundleCreate($bundle, $entity_type_id);
  public function onBundleRename($bundle, $bundle_new, $entity_type_id);
  public function onBundleDelete($bundle, $entity_type_id);

Interface: \Drupal\Core\Field\FieldStorageDefinitionListenerInterface
Class: \Drupal\Core\Field\FieldStorageDefinitionListener
Service: field_storage_definition.listener
Methods:
  public function onFieldStorageDefinitionCreate(FieldStorageDefinitionInterface $storage_definition);
  public function onFieldStorageDefinitionUpdate(FieldStorageDefinitionInterface $storage_definition, FieldStorageDefinitionInterface $original);
  public function onFieldStorageDefinitionDelete(FieldStorageDefinitionInterface $storage_definition);

Interface: \Drupal\Core\Field\FieldDefinitionListenerInterface
Class: \Drupal\Core\Field\FieldDefinitionListener
Service: field_definition.listener
Methods:
  public function onFieldDefinitionCreate(FieldDefinitionInterface $field_definition);
  public function onFieldDefinitionUpdate(FieldDefinitionInterface $field_definition, FieldDefinitionInterface $original);
  public function onFieldDefinitionDelete(FieldDefinitionInterface $field_definition);

Cache Clearing

Note that EntityManagerInter::clearCachedDefinitions() invalidated entity type, field and bundle caches.. When replacing that, make sure to call the required combination of EntityTypeManagerInterface::clearCachedDefinitions(), EntityFieldManagerInterface::clearCachedFieldDefinitions() and EntityTypeBundleInfoInterface::clearCachedBundles(). However, in most cases, a manual cache clear is not necessary. Appropriate caches are automatically cleared when installing a new module, creating a new field or bundle.

Constructor changes

Entity manager has extremely wide usage throughout Drupal Core and contrib modules. All these constructors are being updated to rely on the replacement services instead. Whenever, possible, that is done in a BC compatible way:

1. If only a single new service is needed, typically EntityTypeManager, then it is used as a drop-in replacement as EntityManager also implements all interfaces of the replacement services. A @trigger_error() will eventually be triggered if any method is called on EntityManager instead of the non-deprecated replacement.
2. In case there are multiple replacement services necessary, any extra dependency is added as a new optional argument to the constructor, with a fallback to \Drupal::service().

Any contrib or custom module that extends from service classes, controllers, forms, plugins etc. that has the EntityManager service injected will need to update to the replacement service but can only do so after the parent class has been updated accordingly as well.

Some examples and common base classes are highlighted below.

ContentEntityForm (Drupal 8.6)

\Drupal\Core\Entity\ContentEntityForm now expects an implementation of \Drupal\Core\Entity\EntityRepositoryInterface (entity.repository service) as the first constructor argument instead of EntityManagerInterface. This change is backwards compatible since EntityManager implements both interfaces.