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