For many important tasks such as manipulation and locomotion, robots need to make and break contact with their environment. Although such multi-contact systems are common, they pose a significant challenge when it comes to analysis and control. This thesis exploits the local hybrid structure of such problems and presents scalable and fast algorithmic solutions. First, we present an MPC framework for multi-contact systems. The method is based on the alternating direction method of multipliers (ADMM) and is capable of high-speed reasoning over potential contact events. Then, we focus on utilizing tactile measurements for reactive control, which is very natural yet underexplored in the robotics community. We propose a control framework to design provably stabilizing tactile feedback policies by exploiting the local complementarity structure of contact dynamics. Lastly, inspired by the connection between rectified linear unit (ReLU) activation functions and linear complementarity problems, we present a method to analyze stability of multi-contact systems in feedback with ReLU network controllers.