  We present a set of general results on structural features of the $q$-state
Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature
Boltzmann variable $v$ for various lattice strips of arbitrarily great width
$L_y$ vertices and length $L_x$ vertices, including (i) cyclic and M\"obius
strips of the square and triangular lattice, and (ii) self-dual cyclic strips
of the square lattice. We also present an exact solution for the chromatic
polynomial for the cyclic and M\"obius strips of the square lattice with width
$L_y=5$ (the greatest width for which an exact solution has been obtained so
far for these families). In the $L_x \to \infty$ limit, we calculate the
ground-state degeneracy per site, $W(q)$ and determine the boundary ${\cal B}$
across which $W(q)$ is singular in the complex $q$ plane.
