  We study the effects of screened modified gravity of the $f(R)$, dilaton and
symmetron types on structure formation, from the quasi-linear to the non-linear
regime, using semi-analytical methods. For such models, where the range of the
new scalar field is typically within the Mpc range and below in the
cosmological context, non-linear techniques are required to understand the
deviations of the power spectrum of the matter density contrast compared to the
$\Lambda$-CDM template. This is nowadays commonly tackled using extensive
N-body simulations. Here we present new results combining exact perturbation
theory at the one loop level (and a partial resummation of the perturbative
series) with a halo model. The former allows one to extend the linear
perturbative analysis up to $k\lesssim 0.15{\rm h Mpc}^{-1}$ at the
perturbative level while the latter leads to a reasonable, up to a few percent,
agreement with numerical simulations for $k\lesssim 3{\rm h Mpc}^{-1}$ for
large curvature $f(R)$ models, and $k\lesssim 1{\rm h Mpc}^{-1}$ for dilatons
and symmetrons, at $z=0$. We also discuss how the behaviors of the perturbative
expansions and of the spherical collapse differ for $f(R)$, dilaton, and
symmetron models.
