Contact with the outside world is challenging for robots due to its inherently discontinuous nature — when a foot or hand is touching a surface the forces are completely different than if it is just above the surface. However, most of our computational and analytic tools for planning, learning, and control assume continuous (if not smooth or even linear) systems. Simple models of contact make assumptions (like plasticity and coulomb friction) that are known to not only be wrong physically but also inconsistent. In this talk I will present techniques for overcoming these challenges in order to adapt smooth methods to systems that have changing contact conditions. In particular I will focus on two topics: First, I will show how to use smooth higher-order interpolation to generate new controllers in a contact implicit optimization framework. Second, I will present the “Salted Kalman Filter” for state estimation over hybrid systems.