A system A={a_s+n_s Z}_{s=1}^{k} of k residue classes is called a cover of Z if any integer belongs to one of the k residue classes. This concept was introduced by P. Erdős in the 1950s. Erdős ever conjectured that A is a cover of Z whenever it covers 1,...,2^k.

In this talk we introduce some basic results on covers of Z as well as their elegant proofs. We will also talk about covers of groups by finitely many cosets, give a proof of the Neumann-Tomkinson theorem, and introduce progress on the Herzog-Schőheim conjecture and the speaker's disjoint cosets conjecture.