  We discuss the asymptotics of the eigenvalue counting function for partial
differential operators and related expressions paying the most attention to the
sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl
principal parts and correction terms and asymptotics with non-Weyl principal
parts. Semiclassical microlocal analysis, propagation of singularities and
related dynamics play crucial role.
  We start from the general theory, then consider Schr\"odinger and Dirac
operators with the strong magnetic field and, finally, applications to the
asymptotics of the ground state energy of heavy atoms and molecules with or
without a magnetic field.
