  The multivariate Tutte polynomial (known to physicists as the Potts-model
partition function) can be defined on an arbitrary finite graph G, or more
generally on an arbitrary matroid M, and encodes much important combinatorial
information about the graph (indeed, in the matroid case it encodes the full
structure of the matroid). It contains as a special case the familiar
two-variable Tutte polynomial -- and therefore also its one-variable
specializations such as the chromatic polynomial, the flow polynomial and the
reliability polynomial -- but is considerably more flexible. I begin by giving
an introduction to all these problems, stressing the advantages of working with
the multivariate version. I then discuss some questions concerning the complex
zeros of the multivariate Tutte polynomial, along with their physical
interpretations in statistical mechanics (in connection with the Yang--Lee
approach to phase transitions) and electrical circuit theory. Along the way I
mention numerous open problems. This survey is intended to be understandable to
mathematicians with no prior knowledge of physics.
