All chemical processes exhibit two main universal features. They are stochastic because chemical reactions might happen only after random successful collisions of reacting species, and they are dynamic because the amount of reactants and products changes with time. Since biological processes rely heavily on specific chemical reactions, the stochasticity and dynamics are also crucial features for all living systems. To understand the molecular mechanisms of chemical and biological processes, it is important to develop and apply theoretical methods that fully incorporate the randomness and dynamic nature of these systems. In recent years, there have been significant advances in formulating and exploring such theoretical methods. As an illustration of such developments, in this review the recent applications of stochastic kinetic models for various biological processes are discussed. Specifically, we focus on applying these theoretical approaches to investigate the biological signaling, clearance of bacteria under antibiotics, T cells activation in the immune system, and cancer initiation dynamics. The main advantage of the presented stochastic kinetic models is that they generally can be solved analytically, allowing to clarify the underlying microscopic picture, as well as to explain the existing experimental observations and to make new testable predictions. This theoretical approach becomes a powerful tool in uncovering the molecular mechanisms of complex natural phenomena.