Hamilton cycles in graphs and digraphs

A classical result on Hamilton cycles is Dirac's theorem which states that every graph on n vertices with minimum degree at least n/2 contains a Hamilton cycle. Nash-Williams showed that such a graph even contains many edge-disjoint Hamilton cycle and in 1971 he proposed a conjecture about the maximal number of edge-disjoint Hamilton cycles one can guarantee.
A conjecture of Kelly from 1968 states that every regular tournament can be decomposed into edge-disjoint Hamilton cycles. (A tournament is an orientation of a complete graph.)
I will describe recent results towards these and related conjectures and mention some open problems. The results in this talk will be joint work with Demetres Christofides, Peter Keevash, Fiachra Knox, Richard Mycroft, Deryk Osthus and Andrew Treglown.