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Notifications alert teachers, students and other users about events in Moodle such as new forum posts, assignments needing grading or badges awarded.
New notifications are highlighted with a number in the notifications menu at the top of the screen:
Clicking there displays any new notifications, along with the option to mark all as read (1 below), to go to the notifications preferences page (2) to change how notifications are received, and (3) to view all notifications.
Clicking 'View full notification' displays the notification in a full list, whereas clicking the main body of the notification will take you to the location in Moodle that the event happened, as in this example, to the assignment grading screen.