  Using the results of a cosmological hydrodynamical simulation of the
concordance LambdaCDM model, we study the global properties of the
Sunyaev-Zel'dovich (SZ) effects, both considering the thermal (tSZ) and the
kinetic (kSZ) component. The simulation follows gravitation and gas dynamics
and includes also several physical processes that affect the baryonic
component, like a simple reionization scenario, radiative cooling, star
formation and supernova feedback. Starting from the outputs of the simulation
we create mock maps of the SZ signals due to the large structures of the
Universe integrated in the range 0 < z < 6. We predict that the Compton
y-parameter has an average value of (1.19 +/- 0.32) 10^-6 and is lognormally
distributed in the sky; half of the whole signal comes from z < 1 and about 10
per cent from z > 2. The Doppler b-parameter shows approximately a normal
distribution with vanishing mean value and a standard deviation of 1.6 10^-6,
with a significant contribution from high-redshift (z > 3) gas. We find that
the tSZ effect is expected to dominate the primary CMB anisotropies for l >~
3000 in the Rayleigh-Jeans limit, while interestingly the kSZ effect dominates
at all frequencies at very high multipoles (l >~ 7 10^4). We also analyse the
cross-correlation between the two SZ effects and the soft (0.5-2 keV) X-ray
emission from the intergalactic medium and we obtain a strong correlation
between the three signals, especially between X-ray emission and tSZ effect
(r_l ~ 0.8-0.9) at all angular scales.
