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

In this talk we introduce some basic results on covers of $\mathbb{Z}$ 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-Schöheim conjecture and the speaker's disjoint cosets conjecture.