  We present exact calculations of the partition function of the q-state Potts
model for general q and temperature on strips of the square lattice of width
L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary
conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii)
(FBC_y,TPBC_x)= M\"obius, (iii) (PBC_y,PBC_x)= toroidal, and (iv)
(PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted)
periodic boundary conditions. Results for the L_y=2 torus and Klein bottle
strips are also included. In the infinite-length limit the thermodynamic
properties are discussed and some general results are given for low-temperature
behavior on strips of arbitrarily great width. We determine the submanifold in
the {\mathbb C}^2 space of q and temperature where the free energy is singular
for these strips. Our calculations are also used to compute certain quantities
of graph-theoretic interest.
