  We describe a non-Arrhenius mechanism for slowing down of dynamics that is
inherent to the high dimensionality of the phase space. We show that such a
mechanism is at work both in a family of mean-field spin-glass models without
any domain structure and in the case of ferromagnetic domain growth. The
marginality of spin-glass dynamics, as well as the existence of a `quasi
equilibrium regime' can be understood within this scenario. We discuss the
question of ergodicity in an out-of equilibrium situation.
