  In this thesis, we studied certain mathematical issues related to the
computation of the Chamseddine--Connes spectral action on some fundamental
noncommutative spectral triples, such as the noncommutative torus and the
quantum 3-sphere SUq(2). We showed in particular that a Diophantine condition
on the deformation matrix of the torus is crucial to obtain the spectral action
with real structure. We also studied the question of existence of tadpoles
(linear terms in the gauge potential of the fluctuation of the metric in the
spectral action) for commutative Riemannian geometries, and the construction of
a symbolic global pseudodifferential calculus allowing a generalization of the
Weyl--Moyal product on a Schwartz space of rapidly decaying sections on a
cotangent bundle of a manifold with linearization.
