  We study the deconfinement phase transition in SU(N) gauge theories for
$N$=2,3,4,6,8. The transition is first order for $N \ge 3$, with the strength
increasing as $N$ increases. We extrapolate $T_c/\sqrt{\sigma}$ to the
continuum limit for each $N$, and observe a rapid approach to the large $N$
limit. As $N$ increases the phase transition becomes clear-cut on smaller
spatial volumes, indicating the absence of (non-singular) finite volume
corrections at $N=\infty$ -- reminiscent of large $N$ reduction. The observed
rapid increase of the inter-phase surface tension with $N$ may indicate that
for $N=\infty$ the deconfinement transition cannot, in practise, occur.
