  We examine self-avoiding walks in dimensions 4 to 8 using high-precision
Monte-Carlo simulations up to length N=16384, providing the first such results
in dimensions $d > 4$ on which we concentrate our analysis. We analyse the
scaling behaviour of the partition function and the statistics of
nearest-neighbour contacts, as well as the average geometric size of the walks,
and compare our results to $1/d$-expansions and to excellent rigorous bounds
that exist. In particular, we obtain precise values for the connective
constants, $\mu_5=8.838544(3)$, $\mu_6=10.878094(4)$, $\mu_7=12.902817(3)$,
$\mu_8=14.919257(2)$ and give a revised estimate of $\mu_4=6.774043(5)$. All of
these are by at least one order of magnitude more accurate than those
previously given (from other approaches in $d>4$ and all approaches in $d=4$).
Our results are consistent with most theoretical predictions, though in $d=5$
we find clear evidence of anomalous $N^{-1/2}$-corrections for the scaling of
the geometric size of the walks, which we understand as a non-analytic
correction to scaling of the general form $N^{(4-d)/2}$ (not present in pure
Gaussian random walks).
