In this talk we will discuss the vanishing viscosity problem in a smooth, bounded planar domain. We will recall some known results, including recent work on flows with symmetry, and the well-known Kato criterion. In connection with the vanishing viscosity problem we will discuss vortex sheet solutions of the 2D Euler equations in a smooth, bounded domain. We will conclude with a new criterion for the vanishing viscosity limit to hold, the decay of the so-called vorticity maximal function, valid for non-regular solutions of the 2D Euler equations. This is a report on joint work with P. Constantin, M.C. Lopes Filho and V. Vicol.