Consider an embedding of a graph \(G(v,e)\) with \(v \geq 1\) vertices and \(e \geq 0\) edges into a closed surface s, with r resulting regions. If G is connected and every region is a 2-cell (a so-called 2-cell embedding), Euler’s formula is the relation \(v-e+r=\chi (s),\) where \(\chi (s)\) denotes the Euler characteristic of s. Here we give a generalization of Euler’s formula which applies to any embedding (2-cell or not) of any graph (connected or not) into any surface (orientable or not), with several interesting corollaries. One rather striking corollary is the converse of Euler’s formula itself: If an embedding of a graph \(G(v,e)\) into a closed surface s merely has \(r=e-v+\chi (s)\) regions, the right number for a 2-cell embedding, it is a 2-cell embedding.