The applications of barrier functions to safety analysis, diagnosis, and control have become a popular re-search direction. Barrier functions have been widely applied to solve the safety problems for different types of integer-order systems. This paper starts to discuss the relationship between barrier function and system state-safety for fractional-order nonlinear dynamical systems. First, by using Caputos fractional derivative, this paper revisits the safety theories via barrier functions (BFs). And based on less-zero barrier function, reciprocal barrier function and zeroing barrier function, three extended BFs under the Caputo fractional-order descriptions are designed. Then it establishes a framework by proposing the definitions and criteria of safety, safety-and-stability for a class of fractional-order nonlinear dynamical systems.