Optimization and variational problems typically involve a highly structured blend of smooth and nonsmooth geometry. In nonlinear programming, such structure underlies the design of active-set algorithms, in which a globally convergent process first simplifies the problem by identifying active constraints at the solution; a second phase then employs a rapidly-convergent Newton-type method, with linear models of the simplified problem playing a central role. The theory of partial smoothness formalizes and highlights the fundamental geometry driving identification.'' This dissertation concentrates on the second phase, and understanding accelerated local convergence in partly smooth settings. A key contribution is a simple algorithm for black-box'' nonsmooth optimization, that incorporates second-order information with the usual linear approximation oracle. Motivated by active sets and sequential quadratic programming, a model-based approach is used to prove local quadratic convergence for a broad class of objectives. Promising numerical results on more general functions, as well as simple first-order analogues, are discussed. Beyond optimization, an intuitive linearization scheme for generalized equations is formalized, with simple techniques based on classical differential geometry: manifolds, normal and tangent spaces, and constant rank maps. The approach illuminates fundamental geometric ideas behind active-set acceleration techniques for variational inequalities, as well as second-order theory and algorithms for structured nonsmooth optimization.