  Every element in the first cohomology group of a 3--manifold is dual to
embedded surfaces. The Thurston norm measures the minimal `complexity' of such
surfaces. For instance the Thurston norm of a knot complement determines the
genus of the knot in the 3--sphere. We show that the degrees of twisted
Alexander polynomials give lower bounds on the Thurston norm, generalizing work
of McMullen and Turaev. Our bounds attain their most concise form when
interpreted as the degrees of the Reidemeister torsion of a certain twisted
chain complex. We show that these lower bounds give the correct genus bounds
for all knots with 12 crossings or less, including the Conway knot and the
Kinoshita--Terasaka knot which have trivial Alexander polynomial.
