Landau studied mathematics at the University of Berlin and received his doctorate in 1899 and his habilitation (the post-doctoral qualification required in German universities) in 1901.
Landau taught at the University of Berlin from 1899 until 1909 and held a chair at the University of Göttingen from 1909 until he was forced out by the Nazi regime in 1933. Thereafter he lectured only outside of Germany.
In 1903 Landau gave a much simpler proof than was then known of the prime number theorem and later presented the first systematic treatment of analytic number theory. He also made important contributions to complex analysis.
Hardy wrote that no one was ever more passionately devoted to mathematics than Landau. This is amply evidenced by his books on the axiomatic foundations of analysis and number theory.