Abstract: Research in superconductivity has intensified recently, fueled by the promise of new quantum Majorana particles. Key to this are the symmetries of the superconducting state. In this talk, I will discuss the central role of time-reversal and inversion symmetries in stabilizing superconductivity and the unusual physical properties that arise when these symmetries are broken. The emphasis will be on the nature of the two-electron bound states, that is the Cooper pairs, that underlie superconductivity.