The planar Euler equations describe the motion of a 2-D inviscid incompressible fluid, and also arise as a model problem for the study of the barotropic mode (to put it simply, the vertical average) of the Primitive equations of the ocean. It is a result by Yudovich that, in the space-periodic case, there exist a unique weak solution to the Euler system whenever the initial data has bounded vorticity. Yudovich's result relies on conservation of the sup norm of the vorticity and on endpoint results for the Dirichlet problem which fail in general when the domain is non smooth. By establishing suitable new elliptic regularity results, we construct weak solutions on a planar convex (non-smooth) domain, and prove uniqueness of solutions with bounded vorticity for suitable convex domains with acute corners. We also discuss the related problem of finding a space X between L^infinity and BMO which is both preserved by the Riesz transform (like BMO, unlike L^infinity) and such that the X-norm of the Euler vorticity is conserved (like L^infinity, unlike BMO). This is partly joint work with Roger Temam and Claude Bardos.