  We present in this paper the construction of a pseudodifferential calculus on
smooth non-compact manifolds associated to a globally defined and coordinate
independant complete symbol calculus, that generalizes the standard
pseudodifferential calculus on $\R^n$. We consider the case of manifolds $M$
with linearization in the sense of Bokobza-Haggiag, such that the associated
(abstract) exponential map provides global diffeomorphisms of $M$ with $\R^n$
at any point. Cartan--Hadamard manifolds are special cases of such manifolds.
The abstract exponential map encodes a notion of infinity on the manifold that
allows, modulo some hypothesis of $S_\sigma$-bounded geometry, to define the
Schwartz space of rapidly decaying functions, globally defined Fourier
transformation and classes of symbols with uniform and decaying control over
the $x$ variable. Given a linearization on the manifold with some properties of
control at infinity, we construct symbol maps and $\la$-quantization, explicit
Moyal star-product on the cotangent bundle, and classes of pseudodifferential
operators. We show that these classes are stable under composition, and that
the $\la$-quantization map gives an algebra isomorphism (which depends on the
linearization) between symbols and pseudodifferential operators. We study, in
our setting, $L^2$-continuity and give some examples. We show in particular
that the hyperbolic 2-space $\HH$ has a $S_1$-bounded geometry, allowing the
construction of a global symbol calculus of pseudodifferential operators on
$\S(\HH)$.
