We investigate Dirichlet spaces with superharmonic weights on the open unit disk. Some results on these spaces such as the Alexander-Taylor-Ullman inequality, Carleson measures characterization and explicit formula for reproducing kernels will be explained. We also characterize the boundedness and compactness of composition operators on these spaces. There is a relation with Möbius invariant spaces of analytic functions. What we know and what can be investigated will be discussed. This talk consists of results from the joint work with G. Bao and S. Pouliasis.