Transcatheter aortic valve replacement (TAVR) is a rapidly growing field enabling replacement of diseased aortic valves without the need for open heart surgery. However, due to the nature of the procedure and nonremoval of the diseased tissue, there are rates of complications ranging from tissue rupture and coronary obstruction to paravalvular leak, valve thrombosis, and permanent pacemaker implantation. In recent years, computational modeling has shown a great deal of promise in its capabilities to understand the biomechanical implications of TAVR as well as help preoperatively predict risks inherent to device–patient-specific anatomy biomechanical interaction. This includes intricate replication of stent and leaflet designs and tested and validated simulated deployments with structural and fluid mechanical simulations. This review outlines current biomechanical understanding of device-related complications from TAVR and related predictive strategies using computational modeling. An outlook on future modeling strategies highlighting reduced order modeling which could significantly reduce the high time and cost that are required for computational prediction of TAVR outcomes is presented in this review paper. A summary of current commercial/in-development software is presented in the final section.