Is hook_init() still used? If it is not anymore used, how do I convert code implementing hook_init()?
Perhaps this has changed since this question was last answered. But I think the preferred approach to replacing what is going on in hook_init is to create an event subscriber, and adding to the 'request'. Here is how you do that for those that may find it useful.

Example services.yml

services:
  init_subscriber:
    class:   Drupal\mymodule\EventSubscriber\MyModuleSubscriber
    arguments: ['@current_user'] // &lt;- optional args
    tags:
      - {name: event_subscriber} // &lt;- Required Tag


Then you would implement the EventSubscriberInterface (new file in the src/EventSubscriber directory) and in the implemented method getSubscribedEvents you can do something like...

/**
 * {@inheritdoc}
 */
public static function getSubscribedEvents() {
  $events[KernelEvents::REQUEST][] = array('initializeMyModule');
  return $events;
}


And add the method accordingly

/**
 * MyModule
 *
 * @param \Symfony\Component\HttpKernel\Event\GetResponseEvent $event
 *   The event to process.
 */
public function initializeMyModule(GetResponseEvent $event) {
  $request = $event-&gt;getRequest();   
  ...
}