An introduction to digital stochastic simulations for modeling a variety of physical and chemical processes is presented. Despite the importance of stochastic simulations in chemistry, the prevalence of turn-key software solutions can impose a layer of abstraction between the user and the underlying approach obscuring the methodology being employed. This article introduces the use of random variable generators in simulating physical and chemical processes suitable for use in undergraduate chemistry courses through detailed illustrative examples which include DNA sequences, Brownian motion, diffusion, chemical kinetics, and chain-growth polymerization. The results of the stochastic simulations are also compared to theoretical models. Jupyter notebooks containing Python code for running these simulations are provided in the Supporting Information section.