This chapter explores methods for nonsmooth (i.e., gradient-free) and nonconvex optimization. First, we develop the proximal gradient method, which extends gradient descent to nonsmooth problems where the gradient may not exist. Many optimization problems have a smooth objective, such as the least squares error function, and a nonsmooth regularizer, as in LASSO regression. The proximal gradient is an approach to handle these situations. Next, we explore two algorithms that assume very little structure in the optimization problem: the Nelder-Mead downhill simplex method and simulated annealing. The downhill simplex method uses a simplex (i.e., a polytope) to sample the objective function and take steps that deform the simplex towards better objectives. Simulated annealing replicates the process of annealing, or cooling, to create strong metal alloys. In simulated annealing, the optimization algorithm begins “hot”, exploring the objective function, and gradually “cools” to begin exploiting favorable local minima. We then introduce evolutionary algorithms, including genetic algorithms and genetic programming. Finally, we cover particle swarm optimization and ant colony optimization.