We highlight a fundamental concept pivotal in transitioning stochastic processes from Brownian and Gaussian chains and processes to the domain of singular (containing anomalous) chains and processes. This concept serves as a signature across almost all sections of the textbook. We specifically refer to the heavy/fat tail notion in the probability distribution function (PDF) and the principles of long-range dependence and long memory observed in stationary chains/processes. Our discussion encompasses long-range dependence and long-term memory, examining scenarios (mainly) in the presence of a fat-tailed PDF, particularly heavy-tailed probability distribution. The notion of stationarity of a chain/process goes beyond merely indicating an equilibrium state (as understood in the context of the system’s statistical equilibrium). It enables the consideration of stochastic chains/processes in the presence of drift- and relaxation-type processes, which hold significance in the real world. Stationary chains/processes, distinguished by their simplicity compared to more complex non-stationary counterparts, are extensively studied. This is particularly advantageous as they can effectively describe a broad spectrum of reality, spanning physical, biological, financial, and social domains.