  We use moving mesh hydrodynamical simulations to make maps of
Sunyaev-Zeldovich effect. We present these maps for several cosmological models
and explore their lowest moments. We find that the first moment, the mean
Compton $y$ parameter, is typically between $1-2\times 10^{-6}$ for cluster
abundance normalized models, the lower value corresponding to the high density
models, and scales as $\sigma_8^{3-4}$. Rms fluctuations at 10' scale have an
amplitude $ \Delta T/T \sim 1-3 \times 10^{-6}$ in the Rayleigh-Jeans regime.
The amplitude of the power spectrum strongly depends on the power spectrum
normalization and scales as $\sigma_8^{6-9}$. On smaller scales ($l>1000$) the
spectrum is dominated by halos below $10^{13}M_{\sun}$ and so is sensitive to
the thermal history of galactic halo gas. On larger scales ($l<1000$) the power
spectrum is less sensitive to nongravitational energy injection, but becomes
very noisy, with large field to field variations in the power spectrum caused
by rare bright sources in the maps, which dominate in the spectrum over the
large scale structure correlations. Cross-correlation power spectrum with weak
lensing or projected galaxy distribution is significant and the
cross-correlation coefficient is around 0.5 over a wide range of scales.
Comparison with Press-Schechter predictions gives very good agreement for all
the statistics, indicating that there is no significant contribution to SZ from
non-virialized structures. The one point distribution function shows clear
deviations from gaussianity and can be well-approximated as log-normal on small
scales.
