  We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$.
Our first main theorem is an optimal hitting time result. Consider the random
process which includes the edges of $\mathcal{Q}^n$ according to a uniformly
chosen random ordering. Then, with high probability, as soon as the graph
produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint
Hamilton cycles, for any fixed $k\in\mathbb{N}$. Secondly, we obtain a
perturbation result: if $H\subseteq\mathcal{Q}^n$ satisfies
$\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial
subgraph $\mathcal{Q}^n_p$ of $\mathcal{Q}^n$ with $p\in(0,1]$ fixed, then with
high probability $H\cup\mathcal{Q}^n_p$ contains $k$ edge-disjoint Hamilton
cycles, for any fixed $k\in\mathbb{N}$. In particular, both results resolve a
long standing conjecture, posed e.g. by Bollob\'as, that the threshold
probability for Hamiltonicity in the random binomial subgraph of the hypercube
equals $1/2$. Our techniques also show that, with high probability, for all
fixed $p\in(0,1]$ the graph $\mathcal{Q}^n_p$ contains an almost spanning
cycle. Our methods involve branching processes, the R\"odl nibble, and
absorption.
