  We introduce the general notions of an overconvergent site and a
constructible crystal on an overconvergent site. We show that if $V$ is a
geometric materialization of a locally noetherian formal scheme $X$ over an
analytic space $O$ defined over $\mathbb Q$, then the category of constructible
crystals on $X/O$ is equivalent to the category of constructible modules
endowed with an overconvergent connection on the tube $\,]X[_V$ of $X$ in $V$.
We also show that the cohomology of a constructible crystal is then isomorphic
to the de Rham cohomology of its realization on the tube $\,]X[_V$. This is a
generalization of rigid cohomology. Finally, we prove universal cohomological
descent and universal effective descent with respect to constructible crystals
with respect to the $h$-topology. This encompass flat and proper descent and
generalizes all previous descent results in rigid cohomology.
