  In this paper, we first present an algorithm for computing the Hermite normal
form of pseudo-matrices over Pr\"ufer domains. This algorithm allows us to
provide constructive proofs of the main theoretical results on finitely
presented modules over Pr\"ufer domains and to discuss the resolution of linear
systems. In some sense, we generalize the methodology developed by Henri Cohen
for Dedekind domains. Finally, we present some results over Pr\"ufer domains of
dimension one about the Smith normal form.
