The objective of this dissertation is to develop theoretical and computational tools for the study of qualitative changes in the dynamics of systems with discontinuities, also known as nonsmooth or hybrid dynamical systems, under parameter variations. Accordingly, this dissertation is divided into two parts. The analytical section of this dissertation discusses mathematical tools for the analysis of hybrid dynamical systems and their application to a series of model examples. Specifically, qualitative changes in the system dynamics from a nonimpacting to an impacting motion, referred to as grazing bifurcations, are studied in oscillators where the discontinuities are caused by impacts. Here, the study emphasizes the formulation of conditions for the persistence of a steady state motion in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system. A local analysis based on the discontinuity-mapping approach is employed to derive a normal-form description of the dynamics near a grazing trajectory. Also, the results obtained using the discontinuity-mapping approach and direct numerical integration are found to be in good agreement. It is found that the instabilities caused by the presence of the square-root singularity in the normal-form description affect the grazing bifurcation scenario differently depending on the relative dimensionality of the state space and the steady state motion at the grazing contact. The computational section presents the structure and applications of a software program, TC-HAT, developed to study the bifurcation analysis of hybrid dynamical systems. Here, we present a general boundary value problem (BVP) approach to locate periodic trajectories corresponding to a hybrid dynamical system under parameter variations. A methodology to compute the eigenvalues of periodic trajectories when using the BVP formulation is illustrated using a model example. Finally, bifurcation analysis of four model hybrid dynamical systems is performed using TC-HAT. / Ph. D.