Notifications and messages
These two functions are new in Moodle. You can reach them via the icons speech bubble and bell to the left of your profile picture.
Notifications
The notification function in Moodle informs users about new forum posts, upcoming task submissions or completed test submissions. New notifications are displayed at the top right, to the left of your profile picture via the bell symbol.

By clicking on the bell you can view the notifications, mark all as read (1), switch to the notification settings (2) or see all notifications (3).
Messages
Again on the left side of your profile picture you will find a speech bubble. Please note: The number behind indicates the number of people who sent you a new message, not the number of messages.

Click on the speech bubble to open the message menu. Here you can read the messages, compose a new message (1), mark all as read (2), configure the settings (3), or see all messages (4).