The objective of this paper is to numerically investigate the unsteady cavitating flow around a four-blade inducer, with focus on the cavitation instability and the flow-induced vibration characteristics. In the numerical simulation, the modified rotation/curvature correction turbulence model and the Zwart cavitation model are used for the simulation of the flow field. The tightly coupled algorithm is adopted for the precise prediction of the fluid-structure interaction, including the calculation of the hydrodynamic loads based on the multiphase fluid dynamics and the computation of the structural displacement via the Finite Element Method (FEM). The results showed that good agreement has been obtained between the experimental and numerical results. The fluctuation of cavity volume is the main cause of the change in the head of the inducer, and the backflow vortex cavitation has little effect on that at this flow condition. The backflow vortex cavity develops and rotates with the blades of the inducer, but with a much lower rotational velocity than that of the blades. The flow-induced vibration of the inducer caused by the unsteady cavitating flow mainly manifests as a first-order bending mode. The backflow vortex cavitation has a significant impact on the vibration of both the blades and the guide-water cone. Besides, a cavitation auto-oscillation at the inlet of the inducer has also been detected based on the phase correlation analysis.