We show how to calculate statistical properties of non-Gaussian random fields. We apply this method to determine the mean size and frequency of occurrence of high and low level excursions of the Rayleigh, Maxwell, Chi-squared, lognormal, rectangular and Gumbel type I random fields. These results permit us to calculate the expected size and frequency of fine-scale hotspots and coldspots expected in the microwave background distribution on the sky under the assumption that it possesses non-Gaussian statistics of the above-mentioned types. This generalizes and extends previous studies which confined attention to the simpler case in which the microwave background radiation was assumed to be a Gaussian random field. We also discuss whether it will be possible to determine observationally whether the underlying statistics of the temperature fluctuations in the microwave background are indeed Gaussian as predicted by the standard theory of inflation.