Non-perturbative aspects of the physics of $Sp(2N)$ gauge theories are interesting for phenomenological and theoretical reasons, and little studied so far, particularly in the approach to the large-$N$ limit. We examine the spectrum of glueballs and the string tension of Yang-Mills theories based upon these groups. Glueball masses are calculated numerically with a variational method from Monte-Carlo generated lattice gauge configurations. After taking continuum limits for $N$ = 1, 2, 3 and 4, we extrapolate the results towards large $N$. We compare the resulting spectrum with that of $SU(N)$ gauge theories, both at finite $N$ and as $N$ approaches infinity.