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Laser welding is used in several industrial applications. It can be distinguished between conduction mode and keyhole mode welding, between pulsed wave and cw laser welding and between CO₂-lasers with a wavelength of 10 μm and various laser types of about 1 μm wavelength. A deeper understanding of laser welding allows improving weld quality, process control and process efficiency. It requires a complementary combination of precise modelling and experimental investigations. The here presented review focuses on modelling of laser keyhole welding, for both wavelength regimes. First, the fundamentals of the laser welding process and its physics such as beam propagation, keyhole formation and melt pool dynamics are addressed. The main approaches for modeling energy transfer from laser beam to keyhole surface as well as fluid flow in the material are then discussed. The most relevant publications …