  We study the number $c_n^{(N)}$ of $n$-step self-avoiding walks on the
$N$-dimensional hypercube, and identify an $N$-dependent \emph{connective
constant} $\mu_N$ and amplitude $A_N$ such that $c_n^{(N)}$ is $O(\mu_N^n)$ for
all $n$ and $N$, and is asymptotically $A_N \mu_N^n$ as long as $n\le 2^{pN}$
for any fixed $p< \frac 12$. We refer to the regime $n \ll 2^{N/2}$ as the
\emph{dilute phase}. We discuss conjectures concerning different behaviours of
$c_n^{(N)}$ when $n$ reaches and exceeds $2^{N/2}$, corresponding to a critical
window and a dense phase. In addition, we prove that the connective constant
has an asymptotic expansion to all orders in $N^{-1}$, with integer
coefficients, and we compute the first five coefficients $\mu_N =
N-1-N^{-1}-4N^{-2}-26N^{-3}+O(N^{-4})$. The proofs are based on generating
function and Tauberian methods implemented via the lace expansion, for which an
introductory account is provided.
