We give the survey of the researches dedicated to the statistical stability phenomenon and the physical-mathematical theory of hyper-random phenomena that takes into account the violation of statistical stability. It is presented the study technique of statistical stability, the results of the theoretical and experimental investigations of statistical stability of various processes, the mathematical apparatus of the theory of hyper-random phenomena, the particularities of mathematical statistics of hyperrandom variables (including ones connected with the law of large numbers and the central limit theorem), and the explanation why the accuracy of actual measurements is limited. The description is constructed on the comparison of the theory of hyper-random phenomena with the probability theory.