In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. However, uniqueness within the broader class of solutions with L^p vorticity remains a key unresolved question. In this talk, I will survey recent advances on this problem and present new nonuniqueness results, obtained via the convex integration method. This work is in collaboration with Colombo and Kumar.