In this paper the authors prove the following results (via a unified approach) for all sufficiently large \(n\):

(i) [\(1\)-factorization conjecture] Suppose that \(n\) is even and \(D\geq 2\lceil n/4\rceil -1\). Then every \(D\)-regular graph \(G\) on \(n\) vertices has a decomposition into perfect matchings. Equivalently, \(\chi'(G)=D\).

(ii) [Hamilton decomposition conjecture] Suppose that \(D \ge \lfloor n/2 \rfloor \). Then every \(D\)-regular graph \(G\) on \(n\) vertices has a decomposition into Hamilton cycles and at most one perfect matching.

(iii) [Optimal packings of Hamilton cycles] Suppose that \(G\) is a graph on \(n\) vertices with minimum degree \(\delta\ge n/2\). Then \(G\) contains at least \({\rm reg}_{\rm even}(n,\delta)/2 \ge (n-2)/8\) edge-disjoint Hamilton cycles. Here \({\rm reg}_{\rm even}(n,\delta)\) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on \(n\) vertices with minimum degree \(\delta\).

(i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case \(\delta= \lceil n/2 \rceil\) of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.