  Using cosmological N-body simulations, we study the abundance of local maxima
(peaks) and minima (dips) identified in the smoothed distribution of halos and
dark matter (DM) on scales of $10-100$s Mpcs. The simulations include Gaussian
and local-type $f_{\rm NL}$ non-Gaussian initial conditions. The expression
derived in the literature for the abundance (irrespective of height) of peaks
for Gaussian fields is surprisingly accurate for the evolved halo and DM
density fields for all initial conditions considered. Furthermore, the height
distribution is very well fitted by a log-normal on quasi-linear scales. The
abundance as a function of scale depends on the cosmological parameters ($H_0$
and background matter densities) through the shape of the power spectrum, but
it is insensitive to the clustering amplitude. Further, the abundance in the
smoothed halo distribution is substantially different in the non-Gaussian from
the Gaussian simulations. The interpretation of this effect is straightforward
in terms of the scale dependence of halo bias in non-Gaussian models. The
abundance of extrema extracted from three-dimensional large galaxy redshift
surveys could be a competitive probe of the cosmological parameters and initial
non-Gaussianity. It breaks the degeneracy between $f_{\rm NL}$ and the
clustering amplitude, making it complementary to counts of galaxy clusters and
peaks in weak-lensing maps.
