This paper introduces an algorithm for calculating all discrete point symmetries of a given partial differential equation with a known nontrivial group of Lie point symmetries. The method enables the user to determine the discrete symmetries with little more effort than is used to find the Lie symmetries. It is used to obtain the discrete point symmetries of Burgers' equation, the spherical Burgers' equation, and the Harry–Dym equation. The method can be extended to some types of nonlocal symmetry; we derive the quasi-local discrete symmetries of a system of PDEs from gas dynamics.