In this chapter, we discuss the foundational and advanced aspects of the Markov processes, a cornerstone of stochastic modeling and statistical signal processing. The simplicity and robustness of Markov processes arise from their unique memoryless property: the prediction of future states depends solely on the current state, independent of the sequence of events that led there. When the process evolves over a finite state space and discrete time intervals, it is referred to as a Discrete-Time Markov Chain (DTMC). This chapter introduces the mathematical underpinnings, classifications, dynamic behaviors, and real-world applications of Markov processes, particularly in the context of statistical signal processing.