An operator $T$ acting on a Hilbert space $H$ is said to be a two isometry if $${T^*}^2T^2 -2T^* T+I_H= 0,$$ where $T^*$ denote the adjoint of $T$ and $I_H$ is the identity operator. S. Richter proved that an analytic cyclic two-isometry can be viewed as a shift operator on certain Dirichlet spaces. In this talk, we will present some advances in the study of Dirichlet spaces. We will also discuss several natural open problems related to these spaces, focusing on the description of invariant subspaces. Additionally, we will examine estimates of the reproducing kernel and the concept of capacities associated with Dirichlet spaces.