The leishmaniases comprise a complex of diseases characterised by clinical outcomes that range from self-limiting to chronic, disfiguring and stigmatising, to life-threatening. Diagnostic methods, treatments, and vector and reservoir control options exist, but deciding the most effective interventions requires a quantitative understanding of the population level infection and disease dynamics. The effectiveness of any set of interventions has to be determined within the context of operational conditions, including economic and political commitment. Mathematical models are the best available tools for studying quantitative systems crossing disciplinary spheres (biology, medicine, economics) within environmental and societal constraints.In 2005, the World Health Assembly and government health ministers of India, Nepal, and Bangladesh signed a Memorandum of Understanding to eliminate the life threatening form of leishmaniasis, visceral leishmaniasis (VL), on the Indian subcontinent by 2015 through a combination of early case detection, improved treatments, and vector control. The elimination target is <1/10,000 population at the district or subdistrict level compared to the current 20/10,000 in the regions of highest transmission.Towards this goal, this chapter focuses on mathematical models of VL, and the biology driving those models, to enable realistic predictions of the * Corresponding author Email address: orin.courtenay@warwick.ac.uk (Orin Courtenay)Preprint submitted to Advances in Parasitology July 18, 2016 best combination of interventions. Several key issues will be discussed which have affected previous modelling of VL and the direction future modelling may take. Current understanding of the natural history of disease, immunity (and loss of immunity), as well as stages of infection and their durations are considered particularly for humans, but also for dogs. Asymptomatic and clinical infection are discussed in the context of their relative roles in Leishmania transmission, as well as key components of the parasite-sandfly-vector interaction and intervention strategies including diagnosis, treatment and vector control. Gaps in current biological knowledge and potential avenues to improve model structures and mathematical predictions are identified. Underpinning the marriage between biology and mathematical modelling, the content of this chapter represents the first step towards developing the next generation of models for VL.