This paper deals with Mayer's problem for control systems and differential games with discontinuous terminal cost. There are two main results in the paper. The first one says that the value function for control systems can be characterized as the unique solution in suitable sense to the Hamilton-Jacobi-Bellman equation without any regularity assumptions on the terminal cost. For differential games satisfying Isaacs's minmax condition, the second main result says that the value function is the unique solution to the Hamilton-Jacobi-Isaacs equation when the terminal cost is semicontinuous. This allows to prove the existence of the value under Isaacs's condition. This paper extends some results already well known in the continuous case.