Structure preserving numerical integrators aim to preserve as many of the physical invariants of a dynamical system as possible, since this typically results in a more qualitatively accurate simulation. Computational geometric mechanics is concerned with a class of structured integrators based on discrete analogues of Lagrangian and Hamiltonian mechanics. Discrete theories of exterior calculus and connections on principal bundles provide some of the mathematical foundations of computational geometric mechanics, and address the question of how to obtain canonical discretizations that preserve, at a discrete level, the important properties of the continuous system. Some recent progress on the construction of a combinatorial formulation of discrete exterior calculus based on primal simplicial complexes, and circumcentric dual cell complexes will be presented. These techniques have been used to systematically recover discrete vector differential operators such as the Laplace--Beltrami operator. For discrete connections, the discrete analogue of the Atiyah sequence of a principal bundle is considered, and a splitting of the discrete Atiyah sequence is related to discrete horizontal lifts and discrete connection forms. Continuous connections can be obtained by taking the limit of discrete connections in a natural way. Examples spanning the work on exterior calculus and connections include the discrete Levi-Civita connection for a semidiscrete Riemannian manifold, and the curvature of an abstract simplicial complex endowed with a metric on the vertices.