This work presents an introduction to popular damage and fracture models along with the major breakthroughs and the current state of the art. The theoretical and computational aspects of each model are discussed. While the damage models incorporate the microstructural effects, the fracture models consider the damage mainly at the macro level. The damage models such as Continuum damage mechanics, lattice models, nonlocal damage models, and fracture models such as Extended Finite Element method, Phase field and Peridynamics are presented. Both the discrete and diffused crack approaches along with the numerical solution techniques are presented. A special emphasis on the recently developed Peridynamic models and the different formulations associated with Phase field models is made.