Scheduled list contains tasks that should be done at exact dates.
The list is displayed for each Project and Area and for People. Personal Scheduled list contains tasks assigned to the current user from corresponding Project or Area-level lists.

When you create task in Scheduled list (using Create New Task option), please note an additional “Schedule” filed that is required.

If you move an existing task to Scheduled list using drag-n-drop, the date should be selected in a calendar displayed on the left panel after you drop the task:

The list of Scheduled tasks is grouped by dates, and there are special filters displayed on top of the list:

When the scheduled date comes for a task, it is automatically moved to Today list. Please note, that the date calculation is based on Project or Area time zone (see Project/Area settings).