  We consider the Langevin dynamics of a many-body system of interacting
particles in $d$ dimensions, in a very general setting suitable to model
several out-of-equilibrium situations, such as liquid and glass rheology,
active self-propelled particles, and glassy aging dynamics. The pair
interaction potential is generic, and can be chosen to model colloids, atomic
liquids, and granular materials. In the limit ${d\to\infty}$, we show that the
dynamics can be exactly reduced to a single one-dimensional effective
stochastic equation, with an effective thermal bath described by kernels that
have to be determined self-consistently. We present two complementary
derivations, via a dynamical cavity method and via a path-integral approach.
From the effective stochastic equation, one can compute dynamical observables
such as pressure, shear stress, particle mean-square displacement, and the
associated response function. As an application of our results, we derive
dynamically the `state-following' equations that describe the response of a
glass to quasistatic perturbations, thus bypassing the use of replicas. The
article is written in a modular way, that allows the reader to skip the details
of the derivations and focus on the physical setting and the main results.
