In this talk, we investigate the dimensions of random covering sets. Let $E = \limsup_{n\to\infty}(A_n + x_n)$, where $(A_n)$ is a sequence of Lebesgue measurable sets in the $d$-torus, and $(x_n)$ a sequence of independent random vectors in the $d$-torus with uniform distribution. We determine the almost sure values of the Hausdorff and packing dimensions of $E$ when $A_n$ are open or, more generally, Lebesgue measurable satisfying certain density assumption. The result extends to random covering sets in $\mathbb R^d$ and Riemannian manifolds. This is joint work with Esa Järvenpää, Maarit Järvenpää and Ville Suomala.
