In this talk we look at polynomials having the property that all compositional iterates are irreducible, which we call dynamical irreducible. After surveying some previous results (mostly over finite fields), we will concentrate on the question of the dynamical irreducibility of integer polynomials being preserved in reduction modulo primes. More precisely, for a class of integer polynomials $f$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions, he set of primes $p$ such that $f$ is dynamical irreducible modulo $p$ is of relative density zero. The proof of this result relies on a combination of analytic (the square sieve) and diophantine (finiteness of solutions to certain hyperelliptic equations) tools, which we will briefly describe.