Go to Settings → Updates and click Check for updates.
Turn on Check on startup if you want Burrow to check once every time it starts. It’s off by default.
When a new version is out, the sidebar shows an Update to x.y.z hint. The sidebar also shows the version you’re running.
Install the update
Burrow opens the download in your browser. Install it like the first time: open burrowclient.dmg, drag Burrow Client into Applications, and go through the Privacy & Security step again.
The AppImage downloads the new version and installs it by itself. Before it replaces the app, it checks the download against the SHA-512 hash published with the release, so a broken or altered download is caught.
If you use the rpm, Burrow opens the download in your browser instead.
Your data stays where it is.
What Burrow sends
Burrow only contacts api.github.com, and only when you click Check for updates, or at startup if you turned that on. The request carries nothing about you except your IP address.
Download links in the answer are only used if they point to Burrow’s own releases.