  We introduce and study a remarkable family of real probability measures
$\pi_{st}$, that we call free Bessel laws. These are related to the free
Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and
$\pi_{1t}=\pi^{\boxplus t}$. Our study includes: definition and basic
properties, analytic aspects (supports, atoms, densities), combinatorial
aspects (functional transforms, moments, partitions), and a discussion of the
relation with random matrices and quantum groups.
