  Within the framework of Connes' noncommutative geometry, we define and study
globally non-trivial (or topologically non-trivial) almost-commutative
manifolds. In particular, we focus on those almost-commutative manifolds that
lead to a description of a (classical) gauge theory on the underlying base
manifold. Such an almost-commutative manifold is described in terms of a
'principal module', which we build from a principal fibre bundle and a finite
spectral triple. We also define the purely algebraic notion of 'gauge modules',
and show that this yields a proper subclass of the principal modules. We
describe how a principal module leads to the description of a gauge theory, and
we provide two basic yet illustrative examples.
