Visceral leishmaniasis (VL) is a serious neglected tropical disease that is endemic in 98 countries and presents a significant public health risk. The epidemiology of VL is complex. In the Americas, it is a zoonotic disease that is caused by a parasite and transmitted among humans and dogs through the bite of an infected sand fly vector. The infection also can be transmitted vertically from mother to child during pregnancy. Infected individuals can be classified as asymptomatic or symptomatic; both classes can transmit infection. In part due to its complexity, VL transmission dynamics are not fully understood. Stochastic compartmental epidemic models are a powerful set of tools that can be used to study these transmission dynamics. Past compartmental models for VL have been developed in a deterministic framework to accommodate complexity while remaining computationally tractable. In this work, we propose stochastic compartmental models for VL, which are simpler than their deterministic counterparts, but also have several advantages. Notably, this framework allows us to: (1) define a probability of infection transmission between two individuals, (2) obtain both parameter estimates and corresponding uncertainty measures, and (3) employ formal model comparisons. In this dissertation, we develop both population level and individual level Bayesian compartmental models to study both vector and vertical VL transmission dynamics. As part of this model development, we introduce a compartmental model that allows for two infectious classes. We also derive source specific reproductive numbers to quantify the contributions of different species and infectious classes to maintaining infection in a population. Finally, we propose a formal model comparison method for Bayesian models with high-dimensional discrete parameter spaces. These models, reproductive numbers, and model comparison method are explored in the context of simulations and real VL data from Brazil and the United States.