  We describe statistical and coherence properties of the radiation from x-ray
free electron lasers (XFEL). It is shown that the X-ray FEL radiation before
saturation is described with gaussian statistics. Particularly important is the
case of the optimized X-ray FEL, studied in detail. Applying similarity
techniques to the results of numerical simulations allowed us to find universal
scaling relations for the main characteristics of an X-ray FEL operating in the
saturation regime: efficiency, coherence time and degree of transverse
coherence. We find that with an appropriate normalization of these quantities,
they are functions of only the ratio of the geometrical emittance of the
electron beam to the radiation wavelength. Statistical and coherence properties
of the higher harmonics of the radiation are highlighted as well.
