Features of the Statistical Stability Phenomenon -- The Phenomenon of Statistical Stability and its Properties -- Determinism and Uncertainty -- Formalization of the Statistical Stability Concept -- Dependence of the Statistical Stability of a Stochastic Process on its Spectrum-Correlation Characteristics -- Experimental Study of the Statistical Stability Phenomenon -- Experimental Investigation of the Statistical Stability of Physical Processes over Large Observation Intervals -- Experimental Investigation of the Statistical Stability of Meteorological Data -- Experimental Studies of the Statistical Stability of Radiation from Astrophysical Objects -- Statistical Stability of Different Types of Noise and Process -- The Theory of Hyper-random Phenomena -- Hyper-random Events and Variables -- Hyper-random Functions -- Stationary and Ergodic Hyper-random Functions -- Transformations of Hyper-random Variables and Processes -- Fundamentals of the Statistics of Hyper-random Phenomena -- Principles of the Mathematical Analysis of Divergent and Many-valued Functions -- Divergent Sequences and Functions -- Description of Divergent Sequences and Functions -- Divergent Sequences -- Many-valued Variables, Sequences, and Functions -- Principles of the Mathematical Analysis of Many-valued Functions -- Statistical Laws in Statistical Stability Violation -- The Law of Large Numbers -- The Central Limit Theorem -- Accuracy and Measurement Models -- The Problem of Uncertainty -- Epilogue -- References.