  An important and long-standing open problem in universal algebra asks whether
every finite lattice is isomorphic to the congruence lattice of a finite
algebra. Until this problem is resolved, our understanding of finite algebras
is incomplete, since, given an arbitrary finite algebra, we cannot say whether
there are any restrictions on the shape of its congruence lattice. By a well
known result of Palfy and Pudlak, the problem would be solved if we could prove
the existence of a finite lattice that is not an interval in the lattice of
subgroups of a finite group. Thus the problem of characterizing congruence
lattices of finite algebras is closely related to the problem of characterizing
intervals in subgroup lattices.
  In this work, we review a number of methods for finding a finite algebra with
a given congruence lattice, including searching for intervals in subgroup
lattices. We also consider methods for proving that algebras with a given
congruence lattice exist without actually constructing them. By combining these
well known methods with a new method we have developed, and with much help from
computer software like the UACalc and GAP, we prove that with one possible
exception every lattice with at most seven elements is isomorphic to the
congruence lattice of a finite algebra. As such, we have identified the unique
smallest lattice for which there is no known representation. We examine this
exceptional lattice in detail, and prove results that characterize the class of
algebras that could possibly represent this lattice.
  We conclude with what we feel are the most interesting open questions
surrounding this problem and discuss possibilities for future work.
