In the 1960s, Yudovich proved existence of a unique solution to the two- dimensional incompressible Euler equations when the vorticity, or curl of the velocity, is bounded. In the last few decades, mathematicians have built upon Yudovich’s result by proving existence and uniqueness of solutions with vorticity in various function spaces, such as classical Sobolev spaces and more modern Besov and Triebel-Lizorkin spaces. In this talk, we present some of these results, as well as open problems in this area of research. We also consider extensions of these results to the three-dimensional case under the assumption that the velocity is an axisymmetric vector field.