According to the conservation of mass and momentum conservation principle of the breakdown section in coal-rock system, the cusp catastrophe models of coal and gas outburst are established by using mutation theory, in order to reveal the coal and gas outburst mechanism. The expressions of coal and gas outburst occurring necessary and sufficient conditions are educed, and the occurrence mechanism of coal and gas outburst is analyzed. When the gas pressure in concentrate stress section coal-rock from the frontage tunneling work is ≥ 0.1897 MPa , it meets with the necessary conditions of the heading face mutation to extrude and dump coal and gas. When the gas pressure in concentrate stress section coal-rock from the frontage tunneling work is ≥0.7589MPa, it meets with the necessary conditions of the heading face mutation to burst coal and gas outburst. This research results are useful for analyzing and forecasting coal and gas outburst at locale.