Open Delivery Logs here — find it in the admin under Communications › Delivery Logs.
This is a read-only audit page. It shows the 20 most recent of each of the following.
For every email sent: when it went out, who sent it, the subject, how many recipients it went to, how many were delivered, and how many were not deliverable.
A quick way to keep track of recent push notifications — when each went out, its text, and its status.
For each automatic rule on your site: its type, its subject, the quantity sent so far, the last send time, and whether it's active.
Delivery Logs is a snapshot. For the full history, use these:
Communications › Email Log — every email notification ever queued on your site. Filter by All, Sent, Scheduled or Pending, narrow it to a date range, and search by subject. Click View on a row for the detail.
Communications › Scheduled Pushes — anything queued to send in the future, with a Cancel button so you can stop a push before it goes out.