We present a general, rigorous theory of partition function zeros for lattice
spin models depending on one complex parameter. First, we formulate a set of
natural assumptions which are verified for a large class of spin models in a
companion paper [BBCKK2, math-ph/0304007]. Under these assumptions, we derive
equations whose solutions give the location of the zeros of the partition
function with periodic boundary conditions, up to an error which we prove is
(generically) exponentially small in the linear size of the system. For
asymptotically large systems, the zeros concentrate on phase boundaries which
are simple curves ending in multiple points. For models with an Ising-like
plus-minus symmetry, we also establish a local version of the Lee-Yang Circle
Theorem. This result allows us to control situations when in one region of the
complex plane the zeros lie precisely on the unit circle, while in the
complement of this region the zeros concentrate on less symmetric curves.