Visceral leishmaniasis (VL) is a neglected tropical disease (NTD) caused by Leishmania protozoa that are transmitted by female sand flies. On the Indian subcontinent (ISC), VL is targeted by the World Health Organization (WHO) for elimination as a public health problem by 2020, which is defined as <1 VL case (new and relapse) per 10,000 population at district level in Nepal and sub-district level in Bangladesh and India. WHO is currently in the process of formulating 2030 targets, asking whether to maintain the 2020 target or to modify it, while adding a target of zero mortality among detected cases. The NTD Modelling Consortium has developed various mathematical VL transmission models to gain insight into the transmission dynamics of VL, identify the main knowledge gaps, and predict the feasibility of achieving and sustaining the targets by simulating the impact of varying intervention strategies. According to the models, the current target is feasible at the appropriate district/sub-district level in settings with medium VL endemicities (up to 5 reported VL cases per 10,000 population per year) prior to the start of the interventions. However, in settings with higher pre-control endemicities, additional efforts may be required. We also highlight the risk that those with post-kala-azar dermal leishmaniasis (PKDL) may pose to reaching and sustaining the VL targets, and therefore advocate adding control of PKDL cases to the new 2030 targets. Spatial analyses revealed that local hotspots with high VL incidence remain. We warn that the current target provides a perverse incentive to not detect/report cases as the target is approached, posing a risk for truly achieving elimination as a public health problem although this is taken into consideration by the WHO procedures for validation. Ongoing modelling work focuses on the risk of recrudescence when interventions are relaxed after the elimination target has been achieved.