  The non--linear dynamics of self--gravitating irrotational dust is analyzed
in a general relativistic framework, using synchronous and comoving
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. A general formula for the
gravitational waves generated by the non--linear evolution of cosmological
perturbations is given. It is argued that a stochastic gravitational--wave
background is produced by non--linear cosmic structures, with present--day
closure density $\Omega_{gw} \sim 10^{-5}$ -- $10^{-6}$ on Mpc scale.
