Laser welding of wrought magnesium alloy has been investigated through experimentation and simulation. Laser butt welds and laser lap welds were performed on 2.0 mm thick magnesium alloy AZ31 plates using a 1 kW fiber laser and shielded with argon gas. The effects of laser power and welding speed on weld geometry and microstructure were investigated. Tensile tests were performed to verify weld quality. Through experimentation, a novel processing map was created, which gives the ranges of operating parameters of laser power and welding speed that resulted in viable, defect-free welds. Numerical simulations were performed to predict the weld pool geometry and keyhole stability, and resultant microstructures are shown to be in good agreement with experimental results.