The notion of viscosity solution was introduced in 1980s by Evans and Crandall/Lions, which is one of the most important developments in the  theory of elliptic equations.  It provides a rigorous mathematical framework to describe the correct ``physical" solution of  first or second order PDEs when classical solutions might not exist. Important examples include first order Hamilton-Jacobi equations or second order degenerate elliptic equations (e.g mean curvature type equations) arising from control theory or front propagation problems in real applications.   In this talk, I will go over basic definitions, some important techniques, fundamental results and interesting examples.