Looks like Valve are creating apps to allow users to stream games and video from their Steam libraries to mobile phones/tablets.
Game streaming looks to be limited to the user's local network.
Mobile devices will be getting Steam controller support in some capacity at least.
Hopefully mobile gamers will eventually be able to use the controller to play games like Fortnite, on their phones.
It seems to be unknown if the Steam Video app will be able to stream using Valve's servers, (but probably).
Videos will be available for offline viewing.
Finally, mobile users can discover the creation of Horizon: Zero Dawn.
"During the summer, it plans to release a Steam Video app to stream your Steam video library to Android and iOS devices over Wi-Fi or LTE -- or view it offline."
"A beta will offer only Android support at first, and your device will have to be connected to the local network through either a 5Ghz wireless or wired Ethernet connection. You'll be able to use the Steam Controller, MFI controllers, and other devices to play."
Hopefully both of these apps make their way to the Nintendo Switch, as that device has both hardware video decoding, and built-in controllers (Joycon).