Earlier this year, Sony allowed for universal, OS level button remapping for the DualShock 4 on PlayStation 4. It was a great feature, much appreciated by a lot of people who maybe do not prefer the default control schemes. Microsoft also offered something similar when they released the Xbox One Elite Controller, which offered full button remapping- but this was only restricted to the $150 Elite controller so far.
Microsoft had promised that the button remapping function would be made available for all Xbox One controllers eventually, and today, they have followed up on that– as of today, full button remapping and rebinding is available for all Xbox One controllers (though certain preset button layouts are still restricted to the Elite controller). You can remap your controller by simply going into Settings, and then Ease of Access, or by launching the Xbox Accessories App.
Remapping will work with wired and wireless controllers, and will be functional with third party controllers too. It will be made available for the PC at a later date.