  A state of matter is characterized by its symmetry breaking and elementary
excitations.
  A supersolid is a state which breaks both translational symmetry and internal
$ U(1) $ symmetry.
  Here, we review some past and recent works in phenomenological
Ginsburg-Landau theories, ground state trial wavefunctions and microscopic
numerical calculations. We also write down a new effective supersolid
Hamiltonian on a lattice.
  The eigenstates of the Hamiltonian contains both the ground state
wavefunction and all the excited states (supersolidon) wavefunctions. We
contrast various kinds of supersolids in both continuous systems and on
lattices, both condensed matter and cold atom systems. We provide additional
new insights in studying their order parameters, symmetry breaking patterns,
the excitation spectra and detection methods.
