  The general relativistic non--linear dynamics of a self--gravitating
collisionless fluid with vanishing vorticity is studied in synchronous and
comoving -- i.e. {\em Lagrangian} -- coordinates. Writing the equations in
terms of the metric tensor of the spatial sections orthogonal to the fluid flow
allows an unambiguous expansion in inverse powers of the speed of light. The
Newtonian and post--Newtonian approximations are derived in Lagrangian form;
the non--linear evolution of the system on super--horizon scales, leading to
the so--called ``silent universe", is also briefly discussed. A general formula
for the gravitational waves generated by the non--linear evolution of
cosmological perturbations is given: a stochastic gravitational--wave
background is shown to be produced by non--linear cosmic structures, with
present--day closure density $\Omega_{gw} \sim 10^{-5}$ -- $10^{-6}$ on the
scale of 1 -- 10 Mpc.
