In this paper, we analyze the dynamics of tuned mass dampers with inerters. In the beginning, we describe the influence of inertance value with respect to the overall mass of the damping device. For further analysis, we pick three practically significant cases—each corresponding to different composition of tuned mass damper inertia. Then, we focus on the effects caused by different types of inerters’ nonlinearities. Viscous damping, dry friction and play in the inerter gears influence the dynamics of the tuned mass damper and affect its damping efficiency. Finally, we examine the dynamics of the model that incorporates all of these factors and propose its simplification which is genuine but more convenient. Our results show how to adjust the inerter type and the parameters depending on our needs and intended application. The knowledge on how to model the behavior of tuned mass dampers with inerters will be of practical use to engineers working with mechanical dampers.