In this review, I recall the nature and the inevitability of t he “sign problem” which plagues attempts to simulate lattice QCD at finite baryon density. I pre sent the main approaches used to circumvent the sign problem at small chemical potential. I sketch how one can predict analytically the severity of the sign problem, as well as the numerically accessible range of baryon densities. I review progress towards the determination of t he pseudo-critical temperature Tc(μ), and towards the identification of a possible QCD critical poi nt. Some promising advances with non-standard approaches are reviewed.

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