The proper justification of the normal practice of Monte Carlo integration must be based not on the randomness of the procedure, which is spurious, but on equidistribution properties of the sets of points at which the integrand values are computed. Besides the discrepancy, which it is proposed to call henceforth extreme discrepancy, another concept, that of mean square discrepancy, can be regarded as a measure of the lack of equidistribution of a sequence of points in a multidinensional cube. Determinate upper bounds can be obtained, in terms of either discrepancy, for the absolute value of the error in the computation of the integral. There exist sequences of points yielding, for sufficiently smooth functions, errors of a much smaller order of magnitude than that which is claimed by the Monte Carlo method. In the case of two dimensions, sequences with optimum properties can be generated with the help of Fibonacci numbers. The previous arguments do not apply to domains of integration which cannot be reduced to multidimensional intervals. Difficult questions arising in this connection still await an answer.In view of the widespread and rather successful--at least within its own limitations--use of the Monte Carlo method, it seems essential to look for a mthemtically satisfactory justification of this procedure. The standard argu-. ment is based on the assumption that we use a random source of points. Such an rgument would correspond to reality if, for instance, we borrowed a roulette from Monte Carlo and used it to determine our sequence of points. Of course, there are more practical ways of generating numbers at. rndom, .mostly using radioactivity or radio noise. So far, however, their use has been very limited on account of the technical difficulties connected with . sufficiently rapid generation of digits at random. It is well known that, in general, sequences, of digits described as "pseudorandom" are used instead. Such sequences are perfectly determined, and the results of computations carried out with their help re equally determined. Consequently, it does not make sense to talk, for instance, of the variance of the results of such computations. Alternatively, it might be argued that the sequences in question contain a certain degree of indeterminacy when programmed procedure for obtaining pseudorandom digits is started t place which one could describe, stretching a point, as random; but even accepting this contention, one would find that the resulting distribution of sequences of digits bears little resemblance to that of genuinely random sequences.The sequences used in Monte Carlo computations have often been described as being constructed in a manner suggesting to the uninitiated a random origin, but there is in mathematics no theorem sying anything about sequences which *