Whether operating in a manufacturing plant or assisting within the home, many robotic tasks requires safe and controlled interaction with a complex and changing world. In this talk, I will present our recent progress on learning and control in contact rich settings. In the first segment, I will show how integrating the non-smooth structure of contact dynamics into a learning framework can dramatically improve accuracy and data efficiency when identifying or learning frictional dynamics. Our approach leads to a well-conditioned bilevel optimization problem, avoiding the numerical stiffness and inaccuracies that plague traditional approaches. In the second part of this talk, I will focus on the role of simple, low-dimensional models used in real-time planning for walking robots. Hand-engineered models, typically based in inverted pendulums, are widely used but lead to fundamental limitations on performance. Our recent work, leveraging trajectory optimization within bilevel optimization, to automatically synthesize simple models designed to succeed across a space of tasks. Time-permitting, I will also discuss our work on using bilinear matrix inequalities to leverage tactile feedback within provably stable control policies.