It is possible to follow Business Tasks, Transports or SAP Objects that are of specific interest to you.
This is done via the Follow button in the top right of the Business Task screen within the Web UI.
From here it is possible to choose which events (approvals, rejections etc) that you follow.
It is also possible to say whether you want to receive email notifications as well as seeing updates in the Following notifications screen within the TE Web UI.
It is also possible to edit the Tasks, Transports and SAP Objects you are following via the Configure Followed Items button within the Following area of the notifications screen.