  We present a framework for general relativistic N-body simulations in the
regime of weak gravitational fields. In this approach, Einstein's equations are
expanded in terms of metric perturbations about a Friedmann-Lema\^itre
background, which are assumed to remain small. The metric perturbations
themselves are only kept to linear order, but we keep their first spatial
derivatives to second order and treat their second spatial derivatives as well
as sources of stress-energy fully non-perturbatively. The evolution of matter
is modelled by an N-body ensemble which can consist of free-streaming
nonrelativistic (e.g. cold dark matter) or relativistic particle species (e.g.
cosmic neutrinos), but the framework is fully general and also allows for other
sources of stress-energy, in particular additional relativistic sources like
modified-gravity models or topological defects. We compare our method with the
traditional Newtonian approach and argue that relativistic methods are
conceptually more robust and flexible, at the cost of a moderate increase of
numerical difficulty. However, for a LambdaCDM cosmology, where nonrelativistic
matter is the only source of perturbations, the relativistic corrections are
expected to be small. We quantify this statement by extracting post-Newtonian
estimates from Newtonian N-body simulations.
