The ground state energy of an atom of nuclear charge $Ze$ and in a magnetic field $B$ is evaluated exactly in the asymptotic regime $Z\to \infty $. We present the results of a rigorous analysis that reveals the existence of 5 regions as $Z\to \infty $ : $B\ll Z^{4/3}, B\approx Z^{4/3}, Z^{4/3}\ll B \ll Z^3, B \approx Z^3, B \gg Z^3$. Different regions have different physics and different asymptotic theories. Regions 1,2,3,5 are described exactly by a simple density functional theory, but only in regions 1,2,3 is it of the semi ical Thomas-Fermi form. Region 4 cannot be described exactly by any simple density functional theory ; surprisingly, it can be described by a simple density matrix functional theory, as found after this talk was presented. [There are two more recent references : Pys. Rev. Lett. 69, 749-752 (1992) and Commun. Pure Appl. Math. (in press for the McKean issue).] A surprising conclusion is that although the magnetic field has a profound effect on the atomic energy in regions 2,3,4 and 5, the atom remains spherical (to leading order) in regions 2 and 3.