  This paper deals with the time evolution in the matter era of perturbations
in Friedman-Lemaitre models with arbitrary density parameter $\Omega$, with
either a zero cosmological constant, $\Lambda = 0$, or with a non-zero
cosmological constant in a spatially flat Universe. Unlike the classical
Eulerian approach where the density contrast is expanded in a perturbative
series, this analysis relies instead on a perturbative expansion of particles
trajectories in Lagrangian coordinates. This brings a number of advantages over
the classical analysis. In particular, it enables the description of stronger
density contrasts. Indeed the linear term is the famous Zel'dovich approximate
solution (1970). We present here a systematic and detailed account of this
approach. We give analytical results (or fits to numerical results) up to the
third order. We then proceed to explore the link between the lagrangian
description and statistical measures. We show in particular that Lagrangian
perturbation theory provides a natural framework to compute the effect of
redshift distortions, using the skewness of the density distribution function
as an example. Finally, we show how well the second order theory does as
compared to other approximat- ions in the case of spherically symmetric
perturbations. We also compare this second order approximation and Zel'dovich
solution to N-body simulations in the description of large-scale structure
formation starting from a power law (n=-2) power spectrum of Gaussian
perturbation. We find that second order theory is both simple and powerful.
