  This text contributes to the foundations of the theory of Berkovich spaces
over $\mathbf{Z}$ and rings with similar properties: rings of integers of
number fields, discrete valuation rings, hybrid fields, etc. We focus on three
main themes that had not been investigated so far: category, topology and
cohomology. As regards the category, our main task is to define a well-behaved
notion of morphism. We then have the suitable setting at our disposal to carry
out and study various constructions: products, fiber products, extensions of
scalars, analytification of schemes, etc. On the topological side, we show that
Berkovich spaces over $\mathbf{Z}$ are locally path-connected. The main
ingredient is an analogue of Noether's normalization lemma, that we obtain
after a careful study of finite morphisms. Finally, we prove that open and
closed discs of arbitrary dimension have no higher coherent cohomology. This
allows us to initiate a theory of overconvergent affinoid domains over
$\mathbf{Z}$, where the analogues of Tate's and Kiehl's theorems hold. As a
consequence of our vanishing statements, we obtain a geometric proof of a
Noetherianity result for certain rings of convergent arithmetic power series
(power series with integral coefficients and positive complex radii of
convergence), generalizing a theorem of D. Harbater from the case of a single
variable to arbitrary many.

Algebraic Geometry;Number Theory