A simulation approach is developed to model laser welding processes based on the meshless Smoothed Particle Hydrodynamics (SPH) method. By means of process simulations, fundamental physical processes and effects during laser welding are numerically investigated. The identified significant physical phenomena and implemented modeling aspects include heat transfer, thermoelasticity, thermal expansion, fluid flow in the melt pool, temperature-dependent surface tension, and recoil pressure on the melt due to evaporation. The main focus in the modeling are the phase transitions melting, evaporation, and solidification as well as fluid-structure interaction and laser-material interaction. For the latter aspect, a co-simulation approach is developed by coupling the SPH model with a ray-tracing program that tracks a large number of light rays in the laser beam to deliver accurate predictions of the local absorption of the laser power at the material surface to the SPH model. With the developed SPH model, a variety of welding scenarios can be simulated, ranging from spot welding in the conduction mode regime to deep penetration welding with beam oscillation. The process simulations allow to track the whole welding process and to analyze the influence and sensitivity of certain process parameters on the weld seam. The laser-material interaction proves to be an essential part besides the fluid flow in the melt pool in understanding the formation of process instabilities during the welding process. Different physical phenomena observed in experiments like spiking and moving step-like structures at the capillary front can be replicated in process simulations, showing that the SPH method is indeed well suited for simulating such a complex manufacturing process.