Yudovich established the well-posedness of the two-dimensional incompressible Euler equations for solutions with bounded vorticity. DiPerna and Majda proved the existence of weak solutions with vorticity in L^p ( p > 1). A celebrated open question is whether the uniqueness result can be generalized to solutions with L^p vorticity. In this talk, we resolve this question in negative for some p > 1. To prove nonuniqueness, we devise a new convex integration scheme that employs non-periodic, spatially-anisotropic perturbations, an idea that was inspired by our recent work on the transport equation. To construct the perturbation, we introduce a new family of building blocks based on the Lamb-Chaplygin dipole. This is a joint work with Elia Bruè and Maria Colombo.