  Building on earlier work introducing the notion of "mod-Gaussian" convergence
of sequences of random variables, which arises naturally in Random Matrix
Theory and number theory, we discuss the analogue notion of "mod-Poisson"
convergence. We show in particular how it occurs naturally in analytic number
theory in the classical Erd\H{o}s-K\'ac Theorem. In fact, this case reveals
deep connections and analogies with conjectures concerning the distribution of
L-functions on the critical line, which belong to the mod-Gaussian framework,
and with analogues over finite fields, where it can be seen as a
zero-dimensional version of the Katz-Sarnak philosophy in the large conductor
limit.
