  We perform a series of high-resolution N-body simulations of cosmological
structure formation starting from Gaussian and non-Gaussian initial conditions.
We adopt the best-fitting cosmological parameters of WMAP (3rd- and 5th-year)
and we consider non-Gaussianity of the local type parameterised by 8 different
values of the non-linearity parameter F_NL. Building upon previous work based
on the Gaussian case, we show that, expressed in terms of suitable variables,
the mass function of friends-of-friends haloes is approximately universal (i.e.
independent of redshift, cosmology, and matter transfer function) to good
precision (nearly 10 per cent) also in non-Gaussian scenarios. We provide
fitting formulae for the high-mass end (M>10^13 M_sol/h) of the universal mass
function in terms of F_NL, and we also present a non-universal fit in terms of
both F_NL and z to be used for applications requiring higher accuracy. In the
Gaussian case, we extend our fit to a wider range of halo masses (M>2.4 x 10^10
M_sol/h) and we also provide a consistent fit of the linear halo bias. We show
that, for realistic values of F_NL, the matter power-spectrum in non-Gaussian
cosmologies departs from the Gaussian one by up to 2 per cent on the scales
where the baryonic- oscillation features are imprinted on the 2-point
statistics. We confirm the strong k-dependence of the halo bias on large scales
(k<0.05 h Mpc^-1) which was already detected in previous studies. However, we
find that commonly used parameterisations based on the peak-background split do
not provide an accurate description of our simulations which present extra
dependencies on the wavenumber, the non-linearity parameter and, possibly, the
clustering strength. We provide an accurate fit of the simulation data that can
be used as a benchmark for future determinations of F_NL with galaxy surveys.
