  We present a general, rigorous theory of Lee-Yang zeros for models with
first-order phase transitions that admit convergent contour expansions. We
derive formulas for the positions and the density of the zeros. In particular,
we show that for models without symmetry, the curves on which the zeros lie are
generically not circles, and can have topologically nontrivial features, such
as bifurcation. Our results are illustrated in three models in a complex field:
the low-temperature Ising and Blume-Capel models, and the $q$-state Potts model
for $q$ large enough.
