  In this paper we 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 $\text{reg}_{\text{even}}(n,\delta)$ 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.
