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 difficulty primarily stems from two key factors: 1) the rapid increase in the number of possible ways that a system can move or behave as the number of contacts increase (as a result of the hybrid structure), and 2) the inherent nonlinearities present in the system's dynamics. As a result, for tasks which require initiating contact with the environment, many state-of-the-art methods struggle as the number of contacts increase. Considering the substantial difficulty of multi-contact problems, it's only natural to raise the question: How can we solve such problems? In addressing this query, this thesis directs its attention toward the simplification of multi-contact problems. It does so by concentrating on local hybrid approximations, wherein the non-smooth, hybrid structure is retained, while linearizing the smooth elements within the dynamics to mitigate the complexities arising from nonlinearities. As a result, we focus on local hybrid models called linear complementarity systems which are simple models that qualitatively capture the underlying non smooth, hybrid structure. Employing these local hybrid models, this thesis presents scalable and fast algorithmic solutions for challenging multi-contact problems. First, we present the first real-time MPC framework for multi-contact manipulation. 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 tactile feedback policies for multi-contact systems by exploiting the local complementarity structure of contact dynamics. This framework can close the loop on tactile sensors and it is non-combinatorial, enabling optimization algorithms to automatically synthesize provably stable control policies. Then, 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.