A survey on Hamilton cycles in directed graphs

We survey some recent results on long-standing conjectures regarding Hamilton cycles in directed graphs, oriented graphs and tournaments. We also combine some of these to prove the following approximate result towards Kelly’s conjecture on Hamilton decompositions of regular tournaments: the edges of every regular tournament can be covered by a set of Hamilton cycles which are ‘almost’ edge-disjoint. We also highlight the role that the notion of ‘robust expansion’ plays in several of the proofs. New and old open problems are discussed.