Non-smooth and non-convex models arise in several important applications of mechanics and engineering. The interest in this field is growing from both mathematicians and engineers. The study of numerous industrial applications, including contact phenomena in statics and dynamics or delamination effects in composites, requires the consideration of non-smoothness and non-convexity. The mathematical topics discussed in this book include variational and hemi-variational inequalities, duality, complementarity, variational principles, sensitivity analysis, eigenvalue and resonance problems, and minimax problems. Applications are considered in the following areas among others: non-smooth statics and dynamics, stability of quasi-static evolution processes, friction problems, adhesive contact and debonding, inverse problems, pseudoelastic modelling of phase transitions, chaotic behaviour in nonlinear beams, and nonholonomic mechanical systems. This volume contains 22 chapters written by various leading researchers and presents a cohesive and authoritative overview of recent results and applications in the area of nonsmooth and non-convex mechanics.