We obtain new upper bounds on the minimal density of lattice coverings of R^n by dilates of a convex body K. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice L satisfies L+K=R^n. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem. Joint work with Or Ordentlich and Oded Regev.