Although inertial particle-laden flows occur in a wide range of industrial and natural processes, there is a lack of fundamental understanding of these flows and continuum-level governing equations needed to predict transport and particle distribution. Towards this effort, the Taylor-Couette flow (TCF) system has been used recently to study the flow behavior of particle-laden fluids under inertia. This talk will provide an overview of recent work on the TCF of neutrally buoyant non-Brownian suspensions. The discussion will delve into various studies, including my own contributions, with a particular emphasis on elucidating the impact of finite-sized particles on the series of flow transitions and flow structures. Particles, depending on their size and concentration, cause several significant deviations from Newtonian fluid behavior, including shifting the Reynolds number corresponding to transitions in flow structure and changing the possible structures present in the flow. Furthermore, particles may also migrate depending on the flow structure, leading to hysteretic effects that further complicate the flow behavior. The talk will also address the current state of theoretical and computational modeling efforts to describe existing experimental observations, along with suggestions for potential future directions aimed at enhancing the fundamental understanding of inertial particle-laden flows.