The phenomena of cavitation have been observed in various natural and industrial processes like biomedical engineering and marine engineering. The interaction between a single bubble and free surface, involving the non-spherical collapse of cavitation bubble, nonlinear deformation and instability of free surface, etc., is frontier scientific issues in fluid mechanics and bubble dynamics. In this paper, we summarize research progress and achievements of this field in recent years and discuss typical phenomenon from the perspectives of cavitation dynamics and splash behaviors. For the non-spherical evolution of cavitation bubble, we focus on the key processes such as volume oscillation, jet generation, water hammer effect, collapsing shock wave, and centroid migration based on the dimensionless parameter of kelvin impulse. The mechanism of energy distribution during the cavitation collapse is obtained. For the splash dynamics of free surface, four typical phenomena are summarized based on the generation and development of the thin jet and thick jet: formation of transparent water layer, generation of water column, stable and unstable structure of water crown. Furthermore, we review physical mechanisms on the collapse of cavitation, splash and instability of the free surface by using the theories of Kelvin impulse, Singularity at the curved surface, and Rayleigh-Taylor instability. In addition, we present the effect of curved surfaces on the splash dynamics by investigating the behaviors of free surface of spherical and cylindrical shapes. Finally, remaining challenges and development tendency for future research are given.