We study direct and inverse problems that arise in the vibro-impact oscillations of a discrete system. Specifically, we examine a class of systems with two coordinates undergoing single-or double-sided impacts; however, the presented techniques are sufficiently general to apply to systems with multiple impacts. The analytical methods employed are a nonlinear normal mode (NNM)-type analysis and a boundary value problem (BVP) formulation, and enable the computation of various branches of bifurcating periodic solutions with different impacting characteristics. Additional insight on the dynamics of these systems is obtained by direct integrations of the equations of motion and by numerical Poincare´maps. It is found that the vibro-impact systems considered possess rich nonlinear dynamics, including vibro-impact localized and nonlocalized time-periodic motions, complicated bifurcation structures giving rise to new types of single-and double-sided impacting motions, mode instabilities, and chaotic responses. We also formulate inverse vibro-impact problems, whereby, we seek the class of dynamical systems that produce specified orbits in the configuration plane. The solutions of the inverse problems are generally non-unique, since they can be reduced to underdetermined sets of algebraic equations with multiple infinities of unknowns. Numerical applications are provided to demonstrate the techniques and validate the analytical results.