  Approximations to the exact solutions for gravitational instability in the
expanding Universe are extremely useful for understanding the evolution of
large--scale structure. We report on a series of tests of Newtonian Lagrangian
perturbation schemes using N--body simulations for various power--spectra with
scale--independent indices in the range $-3$ to $+1$. The models have been
evolved deeply into the non--linear regime of structure formation in order to
probe the dynamical and statistical performance of the Lagrangian perturbation
schemes (whose first--order solution contains as a subset the celebrated
``Zel'dovich--approximation'', hereafter ZA). These tests reveal properties of
the approximations at stages beyond the obvious validity of perturbation
theory. Recently, another series of tests of different analytical and
semi--numerical approximations for large--scale structure was conducted with
the result that ZA displays the best dynamical performance in comparison with
the N--body simulations, if the initial data were smoothed before evolving the
model, i.e., a truncated form of ZA (TZA). We show in this contribution that
the excellent performance of TZA can be further improved by going to second
order in the Lagrangian perturbation approach. The truncated second--order
Lagrangian scheme provides a useful improvement over TZA especially for
negative power indices, which suggests it will be very useful for modelling
standard scenarios such as ``Cold--'', ``Hot--'' and ``Mixed--Dark--Matter''.
