Push notifications are a great way to engage app users, as long as it's not annoying. One example of this is wayfinding apps with beacon technology that detect when mobile app users are nearby and send push notifications to let them know something important. In this case, push notifications could alert a user of a special sale, which can be perceived as spammy, or that they have arrived at a certain desired waypoint (more useful, and less spammy), along with interesting information about that location. Think of how useful these are for tourist destinations, like museums, zoos, etc.

You've downloaded xyz zoo app, walk near the big cat area and get a push notice, "grrrrowl, don't miss our feeding time. In 30 minutes you can see us chowing down on fresh antelope quarters".