In this paper, we investigate Dirichlet spaces ${{D}_{\mu }}$ with superharmonic weights induced by positive Borel measures $\mu $ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for ${{D}_{\mu }}$ spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces $H_{\mu }^{2}$ via the balayage of the measure $\mu $ . We show that ${{D}_{\mu }}$ is equal to $H_{\mu }^{2}$ if and only if $\mu $ is a Carleson measure for ${{D}_{\mu }}$ . As an application, we obtain the reproducing kernel of ${{D}_{\mu }}$ when $\mu $ is an infinite sum of point-mass measures. We consider the boundary behavior and innerouter factorization of functions in ${{D}_{\mu }}$ . We also characterize the boundedness and compactness of composition operators on ${{D}_{\mu }}$ .