We present a formalism for continuum and line emission from random clumpy media together with its application to problems of current interest, including CO spectral lines from ensembles of clouds and radio emission from HII regions, supernovae and star-forming regions. For line emission we find that the effects of clump opacity on observed line ratios can be indistinguishable from variations of intrinsic line strengths, adding to the difficulties in determining abundances from line observations. Our formalism is applicable to arbitrary distributions of cloud properties, provided the cloud volume filling factor is small; numerical simulations show it to hold up to filling factors of about 10%. We show that irrespective of the complexity of the cloud ensemble, the radiative effect of clumpiness can be parametrized at each frequency by a single multiplicative correction to the overall optical depth; this multiplier is derived from appropriate averaging over individual cloud properties. Our main finding is that cloud shapes have only a negligible effect on radiation propagation in clumpy media; the results of calculations employing point-like clouds are practically indistinguishable from those for finite-size clouds with arbitrary geometrical shapes.
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