Numerical fluid�structure interaction (FSI) methods have been widely used to predict the cardiac mechanics and associated hemodynamics of native and artificial heart valves (AHVs). Offering a high degree of spatial and temporal resolution, these methods circumvent the need for cardiac surgery to assess the performance of heart valves. Assessment of these FSI methods in terms of accuracy, realistic modeling, and numerical stability is required, which is the objective of this paper. FSI methods could be classified based on how the computational domain is discretized, and on the coupling techniques employed between fluid and structure domains. The grid-based FSI methods could be further classified based on the kinematical description of the computational fluid (blood) grid, being either fixed grid, moving grid, or combined fixed�moving grid methods. The review reveals that fixed grid methods mostly cause imprecise calculations of flow parameters near the blood�leaflet interface. Moving grid methods are more accurate, however they require cumbersome remeshing and smoothing. The combined fixed�moving grid methods overcome the shortcomings of fixed and moving grid methods, but they are computationally expensive. The mesh-free methods have been able to encounter the problems faced by grid-based methods; however, they have been only limitedly applied to heart valve simulations. Among the coupling techniques, explicit partitioned coupling is mostly unstable, however the implicit partitioned coupling not only has the potential to be stable but is also comparatively cheaper. This in-depth review is expected to be helpful for the readers to evaluate the pros and cons of FSI methods for heart valve simulations. © The Author(s) 2021.