This course provides a general overview on the theories of contact and fracture mechanics, relevant for a wide range of disciplines ranging from materials science to engineering and geophysics. Introducing their theoretical foundations, the physical aspects of the resulting nonlinearities induced by such phenomena are emphasized. Numerical methods for their approximate solution are also presented, together with a series of applications to real case studies. The course covers the following topics: I. Contact mechanics A. The Hertzian contact between smooth spheres B. The Cattaneo-Mindlin theory for frictional contact C. Numerical methods for the treatment of the unilateral contact constraint (the penalty method and Lagrange multipliers in FEM, the active set strategy in BEM) D. Contact between rough surfaces: statistical and numerical methods II. Fracture mechanics A. Fundamentals of linear elastic fracture mechanics (LEFM), stress-intensity factors B. Strength and toughness of materials, criteria for crack propagation C. Examples in LEFM solved with the use of the finite element method D. Nonlinear fracture mechanics (NLFM): the cohesive zone model (CZM) E. Numerical implementation of the CZM in the finite element method F. Applications of NLFM to materials science, retrofitting of civil/architectonic structures, composite materials.