  The specific heat of the $x-y$ model is studied on cubic lattices of sizes $L
\times L \times L$ and on lattices $L \times L \times H$ with $L \gg H$ (i.e.
on lattices representing a film geometry) using the Cluster Monte Carlo method.
Periodic boundary conditions were applied in all directions. In the cubic case
we obtained the ratio of the critical exponents $\alpha/\nu$ from the size
dependence of the energy density at the critical temperature $T_{\lambda}$.
Using finite--size scaling theory, we find that while for both geometries our
results scale to universal functions, these functions differ for the different
geometries. We compare our findings to experimental results and results of
renormalization group calculations.
