  We study the growth of structures in modified gravity models where the
Poisson equation and the relationship between the two Newtonian potentials are
modified by explicit functions of space and time. This parameterisation applies
to the $f(R)$ models and more generally to screened modified gravity models. We
investigate the linear and weakly nonlinear regimes using the "standard"
perturbative approach and a resummation technique, while we use the spherical
dynamics to go beyond low-order results. This allows us to estimate the matter
density power spectrum and bispectrum from linear to highly nonlinear scales,
the full probability distribution of the density contrast on weakly nonlinear
scales, and the halo mass function. We analyse the impact of modifications of
gravity on these quantities for a few realistic models. In particular, we find
that the standard one-loop perturbative approach is not sufficiently accurate
to probe these effects on the power spectrum and it is necessary to use
resummation methods even on weakly nonlinear scales which provide the best
observational window for modified gravity as relative deviations from General
Relativity do not grow significantly on smaller scales where theoretical
predictions become increasingly difficult.
