We will talk about Dirichlet spaces $D$ with superharmonic weights, of holomorphic functions on the unit disc and the Moebius invariant spaces $M(D)$ generated by them. We will discuss the relation of the spaces $M(D)$ with classical Moebius invariant spaces and describe some classes of inner functions contained in them. Also, we will describe the isometric composition operators on $BMOA$ equipped with the Moebius invariant $H^p$ norm, $p\in[1,2]$. The talk is partially based on joint works with G. Bao, J. Mashreghi, and H. Wulan.