Gbe a graph with vertex-set V= V( G) and edge-set E= E( G). A 1- factorof G(also called perfect matching) is a factor of Gof degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1- factorizationof Gis a partition of its edge-set Einto 1-factors. For a graph Gto have a 1-factor, | V( G)| must be even, and for a graph Gto admit a 1-factorization, Gmust be regular of degree r, 1 ≤ r≤ | V| − 1.
One can find in the literature at least two extensive surveys  and  and also a whole book  devoted to 1-factorizations of (mainly) complete graphs.
A 1-factorization of
Gis said to be perfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is . In the book  a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.
It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.