Each finite graph on n vertices determines a special (n-1)-fold covering graph that we call TheCover. Several equivalent definitions and basic properties about this remarkable construction are presented. In particular, we show that TheCover of a k-connected graph is k-connected, TheCover of a planar graph is planar and TheCover of a hamiltonian graph is hamiltonian. By studying automorphisms of these covers we also understand the structure of their automorphism groups. A particular nice property is that every automorphism of the base graph lifts to an automorphism of its TheCover. We also show that TheCover of a 3-connected graph X is never a regular cover of X. This is a joint work with Aleksander Malnič and Tomaž Pisanski.