To this point, the PlayStation 4's novel Remote Play function was only accessible on the PlayStation Vita, PlayStation TV, and certain Xperia mobile phones. That's no longer the case, thanks to an unofficial port that lets the official PS4 Remote Play app work on practically any modern Android device.
A small team of developers led by a coder with the handle TheScriptKitty posted the port back in November, but it's starting to get wider notice now that it has been unearthed by the reddit Destiny community. Running the port requires sideloading an APK onto your device, but once that's done, the process of finding and connecting to a local PS4 seems pretty seamless, based on a recent video tutorial. After that initial syncing, the Remote Play connection will even work across the Internet, as long as you leave your PS4 in Rest Mode.
Reddit and XDA users report good results with a number of different phones and Android versions, though lag seems to be a problem, depending on the connections being used (Wi-Fi is definitely recommended over eating through your mobile data plan) The app comes with built-in touchscreen facsimile of the PS4 controller, but you can connect a DualShock 3 or DualShock 4 controller to the phone using a separate app (which requires a rooted version of Android to use).