Let L be a lattice and let C be any subset in R^d. The covering radius of L with respect to C is the infimum over all r > 0 such that L + rC = R^d is. It was conjectured by Minkowski that if C is the set of all x satisfying |x_1 \cdots x_d| \leq 1, then the covering radius of any unimodular lattice L with respect to C is at most 2^{-d}, and this upper bound is obtained if and only if L is in AZ^d, where A is the group of diagonal matrices. In this talk I will discuss covering radii in the positive characteristic setting. In particular, I will talk about the surprising connections between successive minima and covering radii with respect to convex sets, and the solution of the positive characteristic analogue of Minkowski’s conjecture.