We consider Wigner functions in two space dimensions. In particular, we focus on Wigner functions corresponding to energy eigenstates of a non-relativistic particle moving in two dimensions in the absence of a potential. With the help of the Weyl-Wigner correspondence we first transform the eigenvalue equations for energy and angular momentum into phase space. As a result we arrive at partial differential equations in phase space which determine the corresponding Wigner function. We then solve the resulting equations using appropriate coordinates.