Let \(\mathcal {G}\) be the set of all the planar embeddings of a (not necessarily connected) n-vertex graph G. We present a bijection \(\varPhi \) from \(\mathcal {G}\) to the natural numbers in the interval \([0 \dots |\mathcal {G}| - 1]\) . Given a planar embedding \(\mathcal {E}\) of G, we show that \(\varPhi (\mathcal {E})\) can be decomposed into a sequence of O(n) natural numbers each describing a specific feature of  \(\mathcal {E}\) . The function \(\varPhi \) , which is a ranking function for \(\mathcal {G}\) , can be computed in O(n) time, while its inverse unranking function \(\varPhi ^{-1}\) can be computed in \(O(n \alpha (n))\) time. The results of this paper can be practically applied to uniformly generating the planar embeddings of a graph G at random or to enumerating such embeddings with an amortized constant delay. Additionally, they can be used for counting, enumerating, or uniformly generating constrained planar embeddings of G.