Only compulsory for updates... the rest can be done via Bluetooth. Even so, it's a good idea to use the robot for notifications, to make it easier without having to go near it via Bluetooth. But at home I don't have wifi everywhere, and it works very well. When it crashes, however, it doesn't send a notification and you sometimes have to look for it.
I was only able to get the Mova up and running after I had wifi on my mobile. I tried it with a hotspot, but it didn't work. Now it runs without Wi-Fi and is operated via Bluetooth. This means that operation is somewhat limited, as you always have to be nearby...