  We study the off-equilibrium dynamics of a particle in a general
$N$-dimensional random potential when $N \to \infty$. We demonstrate the
existence of two asymptotic time regimes: {\it i.} stationary dynamics, {\it
ii.} slow aging dynamics with violation of equilibrium theorems. We derive the
equations obeyed by the slowly varying part of the two-times correlation and
response functions and obtain an analytical solution of these equations. For
short-range correlated potentials we find that: {\it i.} the scaling function
is non analytic at similar times and this behaviour crosses over to
ultrametricity when the correlations become long range, {\it ii.} aging
dynamics persists in the limit of zero confining mass with universal features
for widely separated times. We compare with the numerical solution to the
dynamical equations and generalize the dynamical equations to finite $N$ by
extending the variational method to the dynamics.
