[iOS] Controller support for MFi certified devices is now available. For players using iOS 13, PlayStation 4 and Xbox One controllers will also work.

[Android] Controller support is available on some Android devices. Not all brands will work. For example, Steelseries Nimbus will not work due to incorrect button mappings (Steelseries Stratus XL, which is marketed as an Android controller, should work however).

Compatibility also depends on your Android OS version and brand of your device. You should do research into which controllers will work correctly for your OS and device, but looking for Android compatible controllers is a good start.

PS4 and Xbox One controller support is generally pretty good on Android 9+. There is native controller support in Android 10.

Here are the controls:

Left stick - Move
Right stick - Camera
A (X) - Tap to jump, hold to boost
B (Ο) - Chirp / shout / call lost lights
X (⚿) or Right Trigger (R2) - Activate nearest hint icon (light candle, meditate, etc)
Y (Δ) - Toggle expression menu
Left Shoulder Button (L1) - Press to view currency and wing buff status. Press again to access purchase menu.
Right Shoulder Button (R1) - Toggle flight mode when in air
Left Trigger (L2) - Hold to activate stepping mode while on ground
Menu/pause button (Options) - Main Menu

Menu mode:
Right stick or D-pad - Scroll through menu items
A (X) - Select item
Left/Right Shoulder Button (L1/R1) - Previous / Next Page
B (Ο)- Back / Exit

You can also invert the camera controls as well as the flight controls in the same menu.