We consider the interaction between an incompressible, viscous fluid and a poroelastic medium modelled by a Biot equation. Both linear and nonlinear cases are considered. In the linear case, the fluid flow and the poroelastic structure are coupled across a fixed interface, while in the nonlinear case, they are coupled via the moving interface. We discuss the existence of weak solutions for both cases and highlight the primary mathematical challenges associated with the analysis of fluid-poroelastic structure interactions (FPSI), particularly focusing on the difficulties arising from studying the moving boundary problem. Furthermore, we explore numerical methods for solving FPSI problems. The presented results are joint work with L. Bociu, M. Bukač, S. Čanić, J. Kuan and J. Webster.