How do I upgrade my Apple Time Capsule firmware?

As with many network devices, it can indeed be puzzling to figure out how to upgrade the firmware / software on the Apple Time Capsule, and your first guess, either the Time Machine software or Time Machine control panel, is not where you need to go. Neither seems to have much clue about the Time Capsule, per se.

(for those of you that aren't sure which is which, the Time Capsule is a physical device, a network-accessible hard drive that serves as the target for computer backups done with the Time Machine software)

Turns out that the key piece of software for updating the Time Capsule is (ready for this?) Apple's Airport Utility. Why? Because the Time Capsule is also a wireless Airport base station. Ah, so that makes some sense after all.

If there's an update available, when you start up this utility, which you can find in Applications --> Utilities --> Airport Utility, you'll see this:

Since you want to update your Time Capsule device, click "Update".

Shortly you'll find that you've switched over to "Software Update", which will check for new software for everything Apple on your Mac:

Sure enough, there is an update for the Airport system:

Proceed and Software Update will dutifully download everything it needs:

When it's done downloading - and probably rebooting your computer - you need to relaunch Airport Utility. This time it'll show up a bit differently:

Since you'll be changing your network connection, it'll warn you that you'll go offline once the process starts:

That's okay. Go for it...

Once it's updated, the Time Capsule needs to reboot:

and in a short while:

Success. Now you know how to update your Apple Time Capsule! :-)