Unknown input observers (UIOs) are observers that have stable error dynamics that are independent of unknown inputs. This paper studies such observers for non-linear systems, and shows that the error dynamics for a non-linear UIO has the same structure as the error dynamics of a non-linear observer without unknown inputs. This result is first used to provide synthesis inequalities for UIOs for a class of non-linear systems, and secondly, to inspire the design of an observer for estimation of vehicle lateral velocity on banked roads.