  We present exact calculations of the Potts model partition function
$Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex
square-lattice strip graphs $G$ for a variety of transverse widths $L_t$ and
for arbitrarily great length $L_\ell$, with free longitudinal boundary
conditions and free and periodic transverse boundary conditions. These have the
form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}}
c_{Z,G,j}(\lambda_{Z,G,j})^{L_\ell}$. We give general formulas for $N_{Z,G,j}$
and its specialization to $v=-1$ for arbitrary $L_t$ for both types of boundary
conditions, as well as other general structural results on $Z$. The free energy
is calculated exactly for the infinite-length limit of the graphs, and the
thermodynamics is discussed. It is shown how the internal energy calculated for
the case of cylindrical boundary conditions is connected with critical
quantities for the Potts model on the infinite square lattice. Considering the
full generalization to arbitrary complex $q$ and $v$, we determine the singular
locus ${\cal B}$, arising as the accumulation set of partition function zeros
as $L_\ell \to \infty$, in the $q$ plane for fixed $v$ and in the $v$ plane for
fixed $q$.
