The notion of randomness has always been rather perplexing. Although it is frequently used in natural and social science, both technically and informally, it seems to have been somewhat neglected by philosophers of science ever since the discussion of the foundations of the so-called frequency theory of probability, in which it was assigned a basic role, has faded. Yet this discussion is of such significance that any attempt at clarifying the notion of randomness will have to relate to it. After a few preliminary remarks on some of the problems and puzzles of randomness, I shall, therefore, expound and discuss a concept of random distribution of a property in classes and sequences, defined in terms of relative frequencies and their limits. Because of certain shortcomings of this concept it appears advisable to turn to probabilities, in terms of which a quite different concept, viz., that of random conjunction of properties, can readily be defined as stochastic independence. This concept still has features clashing with the ordinary sense of ‘randomness’ which become manifest in cases where certain probabilities assume extreme values. However, when we take measures defined in information theory as measuring the degree of randomness, to which purpose they lend themselves particularly well, we find that these seemingly troublesome cases are rather harmless. A by-product of the discussion of measures of randomness is the concept of primitive randomness. The conclusion points out some further problems.