Although the general principles of Monte Carlo and other simulation techniques have been known since the turn of the century, lack of efficient computational facilities has restricted their general application. The rapid advances in the field of electronic computing during the last three decades, however, have produced a new awareness of the potential of such techniques and as computing becomes even more sophisticated, such methods will no doubt play an increasingly important role in future scientific investigation. -- Effective application of Monte Carlo methods requires access to long sequences of random numbers. Since perfectly random numbers can not, of course, be obtained by practical means, all sets produced to date must properly be termed "pseudo-random". It is generally accepted, in the published literature, that such sets will be more limited in their application than perfectly random sets would be. Even though considerable research has gone into producing sequences for general application, such sequences produced to date are not equally satisfactory for all purposes and must be considered in the light of the particular problem under investigation. -- In this thesis, we consider the problem of finding sequences suitable for the Monte Carlo calculation of definite integrals. After a particular sequence is generated and tested for randomness, it is used in the evaluation of three definite integrals. The results of the statistical tests for each sequence are then compared with the values of the integrals produced by that sequence and an attempt is made to determine which properties a sequence should possess in order to produce good results in this application. Throughout the thesis, several small innovations are introduced which, we believe, have not been reported by other authors
