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• A general Filippov-type predator-prey model with a refuge is proposed. • A The necessary and sufficient conditions for the global asymptotical stability of a standard cycle, a touching cycle and a sliding cycle are present. • The sliding cycle is proved to be globally finite-time stable. • Several kinds of sliding bifurcations are investigated. In this paper, a general Filippov-type predator-prey model with a refuge is presented. We propose a discontinuous predator-prey model incorporating a threshold policy by extending a general continuous predator-prey model. By employing the qualitative analysis theory related to Filippov systems, the necessary and sufficient conditions for the global asymptotical stability of a standard cycle, a touching cycle and a sliding cycle are obtained respectively. Furthermore, the sliding cycle is globally finite-time stable. Especially, several kinds of sliding bifurcations including boundary node bifurcation, boundary focus bifurcation and grazing bifurcation are studied. Moreover, two specific models are provided to verify the main results obtained from the general model. [ABSTRACT FROM AUTHOR]