  Motivated in part by the pseudo-Nambu Goldstone Boson mechanism of
electroweak symmetry breaking in Composite Higgs Models, in part by dark matter
scenarios with strongly coupled origin, as well as by general theoretical
considerations related to the large-N extrapolation, we perform lattice studies
of the Yang-Mills theories with $Sp(2N)$ gauge groups. We measure the string
tension and the mass spectrum of glueballs, extracted from appropriate 2-point
correlation functions of operators organised as irreducible representations of
the octahedral symmetry group. We perform the continuum extrapolation and study
the magnitude of finite-size effects, showing that they are negligible in our
calculation. We present new numerical results for $N=1$, $2$, $3$, $4$, combine
them with data previously obtained for $N=2$, and extrapolate towards
$N\rightarrow \infty$. We confirm explicitly the expectation that, as already
known for $N=1,2$ also for $N=3,4$ a confining potential rising linearly with
the distance binds a static quark to its antiquark. We compare our results to
the existing literature on other gauge groups, with particular attention
devoted to the large-$N$ limit. We find agreement with the known values of the
mass of the $0^{++}$, $0^{++*}$ and $2^{++}$ glueballs obtained taking the
large-$N$ limit in the $SU(N)$ groups. In addition, we determine for the first
time the mass of some heavier glueball states at finite $N$ in $Sp(2N)$ and
extrapolate the results towards $N \rightarrow +\infty$ taking the limit in the
latter groups. Since the large-$N$ limit of $Sp(2N)$ is the same as in $SU(N)$,
our results are relevant also for the study of QCD-like theories.
