Does Pushover use SMS/Text Messages?

For most users, no. Pushover does all of its communication with your device over the Internet, using the connection your phone already has to Google or Apple for receiving push notifications.

For Pushover for Teams users, your account can be configured to enable SMS backup delivery for priority 2 messages.

While you will not incur per-message SMS charges for receiving Pushover notifications, it also means that Pushover requires a working WiFi or cellular data connection to receive notifications. If your phone is out of range of a data provider, it will receive all queued up notifications once it establishes a working data connection.