The stability and bifurcation of a predator-prey system with Holling type Ⅲ functional response at fixed point are analyzed. The fixed point classification of the system is discussed. When the fixed point is hyperbolic, the relationship between the modulus of the root of the characteristic equation and the size of 1 is analyzed. When the fixed point is non-hyperbolic, the center manifold theory and bifurcation theory are used to analyze the eigenvalues. The stability criterion is obtained when the fixed point of the system is hyperbolic. When the fixed point is non-hyperbolic, taking the positive fixed point as an example, Flip bifurcation and Neimark-Sacker bifurcation occur at the positive fixed point of the system, but no transcritical bifurcation occurs. The stability criterion of the periodic orbit or invariant curve bifurcated from the positive fixed point of the system is also obtained. Numerical simulation verifies the correctness of the theoretical analysis. The research results advance the related work in the existing literature.