  We review a gravitational model based on noncommutative geometry and the
spectral action principle. The space-time geometry is described by the tensor
product of a four-dimensional Riemanian manifold by a discrete noncommutative
space consisting of only two points. With a specific choice of the finite
dimensional involutive algebra, the noncommutative spectral action leads to the
standard model of electroweak and strong interactions minimally coupled to
Einstein and Weyl gravity. We present the main mathematical ingredients of this
model and discuss their physical implications. We argue that the doubling of
the algebra is intimately related to dissipation and the gauge field structure.
We then show how this noncommutative spectral geometry model, a purely
classical construction, carries implicit in the doubling of the algebra the
seeds of quantization. After a short review on the phenomenological
consequences of this geometric model as an approach to unification, we discuss
some of its cosmological consequences. In particular, we study deviations of
the Friedmann equation, propagation of gravitational waves, and investigate
whether any of the scalar fields in this model could play the role of the
inflaton.
