Lax has made great contributions to both pure mathematics and applied mathematics. The main research fields are partial differential equations, numerical analysis and calculation, scattering theory, functional analysis and fluid mechanics. His solution to singular integral operator and Cauchy problem with oscillating initial value shows that pseudo-differential operator and Fourier integral operator will play a role in the future. Together with Phillips, he opened up new prospects for the study of scattering theory. Lax is the authority of nonlinear hyperbolic equation and shock wave theory. He has made a decisive contribution to Riemann problem of hyperbolic conservation law equation, Lax shock wave condition and the role of entropy in shock wave theory, and influenced Glimm to get the global solution of nonlinear hyperbolic equation. Then Glimm and Lax obtained the conditions of shock wave formation and attenuation. His conception of KdV equation greatly promoted the theory of completely integrable equation and its connection with other fields. Lax and Levermore also give strict results of small dispersion limit. Lax-Wendroff scheme is the starting point in the numerical method for solving hyperbolic equations. Lax equivalence theorem and approximate stability results. His early Lax-Milgram theorem is one of the basic theorems of linear functional analysis and has important applications.