Quasi-degenerate “pseudospin” doublets were discovered many years ago in both spherical and deformed nuclei [1-5].We show that pseudospin symmetry is an SU(2) symmetry of the Dirac Hamiltonian which occurs when the scalar and vector potentials are opposite in sign but equal in magnitude [6,7].This symmetry occurs independent of the shape of the nucleus: spherical,axial deformed,triaxial,and gamma unstable.We survey the evidence that pseudospin symmetry is approximately conserved for a Dirac Hamiltonian with realistic scalar and vector potentials by examining the energy spectra,the lower components of the Dirac eigenfunctions,the magnetic dipole and Gamow-Teller transitions in nuclei,the upper components of the Dirac eigenfunctions,and nucleon –nucleus scattering [8-14].We shall also search for a fundamental rationale for pseudospin symmetry in terms of chiral symmetry breaking as suggested by QCD sum rules [15].A starting point is an investigation of pseudospin breaking in the nucleon-nucleon interaction by studying nucleon-nucleon scattering as a function of energy [16].Furthermore we show that spin symmetry is an SU(2) symmetry of the Dirac Hamiltonian which occurs when the scalar and vector potentials are equal in magnitude.We show that heavy mesons and anti-nucleons in a nuclear environment have spin symmetry.We then show that the relativistic harmonic oscillator with spin symmetry has an U(3) symmetry and that the relativistic harmonic oscillator with pseudospin symmetry has a pseudo-U(3) symmetry and we derive the generators for both limits [17].