Abstract: A mathematical model of laser keyhole welding, which combines the effects of three dimensional transient keyhole and transient weld pool, is proposed. The coupled effect of laser keyhole welding process is considered with the model based on the multiple reflections Fresnel absorptions, the jump boundary conditions of the keyhole wall, the radiation heat transfer of vapor plume/plasma, the Marangoni force, the friction force in the solid-liquid mixture phase, the buoyancy force, the viscous force, the evaporation latent and the solidification/melting latent, and the conduction, convection and radiation heat transfer of the metal alloys. A Level Set method and a new Fast Sweeping method are developed to solve the transient keyhole evolution equations. A SOLA method is proposed to solve the three-dimensional heat transfer and fluid flow of the transient weld pool. Numerical examples demonstrate that the presented model can be used to reasonably simulate the transient keyhole behaviors and the transient weld pool dynamics during laser keyhole welding.