  We review the current state of the theory of large scale structure in a warm
dark matter (WDM) cosmological model. In particular, we focus on the non-linear
modelling of the matter power spectrum and on the mass function of dark matter
haloes. We describe the results of N-body simulations with WDM and mention the
effects that could be induced by baryonic physics. We also examine the halo
model of large scale structure and its recently suggested modifications for a
WDM cosmology, which account for the small scale smoothness of the initial
matter density field and better fit the results of N-body simulations. Having
described the theoretical models, we discuss the current lower limits on the
WDM particle mass, m_w, which correspond to upper limits on the WDM temperature
under the assumption that the particles are thermal relics. The best such
constraints come from the Ly{\alpha} forest and exclude all masses below 3.3
keV at the 2{\sigma} confidence level. We finally review the forecasts for
future lensing surveys, which will be of the same order of magnitude as the
already existing constraints from the Ly{\alpha} forest data but explore a
different redshift regime.
