Abstract: Anyons are quantum particles with statistics intermediate between bosons and fermions, arising in less than 3 dimensions. There are two possible approaches: one uses multi-valued wave functions, which means topologically non trivial configuration spaces, the other one allows to work in a flat, topologically trivial space, but the non interacting Hamiltonian in this case contains a singular Aharonov-Bohm - type potential. Our analysis lies in the latter approach. The Aharonov-Bohm potential has a singular behaviour at the overlapping configurations. Thus, it is not obvious how to rigorously realize the corresponding Schroedinger operator(s). We mainly focuses on the case of two anyons, which reduces to a one-body problem. The self-adjointness question is investigated and solved via quadratic form techniques. The Dirichlet forms are defined both on the magnetic Gauge and on an appropriate weighted Sobolev space. We also derive the extensions as suitable limits of scaled regular potentials. We present some perspectives for the many-body problem.