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Electronepositron pair production due to the decay of vacuum in ultrastrong laser fields is an interesting topic which is revived recently because of the rapid development of current laser technology. The theoretical and numerical research progress of this challenging topic is reviewed. Many new findings are presented by different approaches such as the worldline instantons, the S-matrix theory, the kinetic method by solving the quantum Vlasov equation or/and the real-time Dirac-Heisenberg-Wigner formalism, the computational quantum field theory by solving the Dirac equation and so on. In particular, the effects of electric field polarizations on pair production are unveiled with different patterns of created momentum spectra. The effects of polarizations on the number density of created particles and the nonperturbative signatures of multiphoton process are also presented. The competitive interplay between the multiphoton process and nonperturbation process plays a key role in these new findings. These newly discovered phenomena are valuable to deepen the understanding of pair production in complex fields and even have an implication to the study of strong-field ionization. More recent studies on the pair production in complex fields as well as beyond laser fields are briefly presented in the view point of perspective future.