  Linked cluster expansions are generalized from an infinite to a finite
volume. They are performed to 20th order in the expansion parameter to approach
the critical region from the symmetric phase. A new criterion is proposed to
distinguish 1st from 2nd order transitions within a finite size scaling
analysis. The criterion applies also to other methods for investigating the
phase structure such as Monte Carlo simulations. Our computational tools are
illustrated at the example of scalar O(N) models with four and six-point
couplings for $N=1$ and $N=4$ in three dimensions. It is shown how to localize
the tricritical line in these models. We indicate some further applications of
our methods to the electroweak transition as well as to models for
superconductivity.
