Landau requires no introduction. He is too well known, although not quite as well understood. Several generations of theoretical physicists learned their trade by struggling through the 10 volumes of his famous Course of Theoretical Physics - known colloquially as Landau and Lifshitz - which Landau supervised, although he did not write a single word. Several dozen physicists - some absolutely first-class - continued to identify themselves as members of the Landau school long after they became scientists in their own right.
About a dozen landmark results in physics bear his name. If one were to choose his most important breakthroughs, these would probably be his theory of phase transitions (1937), the theory of superfluidity in liquid helium (1941), the Ginzburg-Landau phenomenological theory of superconductivity (1950) and the Landau "Fermi-liquid" theory (1956). But his fame is certainly much greater than these concrete results alone might suggest - even if one were to include other formulae, such as the Landau diamagnetism of free electrons (1930) or Landau damping in plasmas (1946).
Landau's published papers are so laconic - consisting mainly of formulae and a few categorical statements on what is right and what is wrong, without sufficient explanation - that even professional physicists often found it hard to understand where his ideas came from.