We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, is applicable on meshes on the manifold with generic elements, and has the same cohomology as the continuous de Rham complex. Notions of local (full and trimmed) polynomial spaces are developed, with compatibility requirements between polynomials on mesh entities of various dimensions. Explicit examples of polynomials spaces are presented. The discrete de Rham complex is then used to set up a scheme for the Maxwell equations on a 2D manifold without boundary, and we show that a natural discrete version of the constraint linking the electric field and the electric charge density is satisfied. Numerical examples are provided on the sphere and the torus, based on a bespoke analytical solution and mesh design on each manifold.