The discovery of the high-temperature superconductors in the 1980’s revealed the existence of materials suspected to have unconventional pairing symmetry characterized by anisotropies in the magnitude and/or phase of the order parameter, in sharp contrast to conventional BCS superconductors.  In this talk, I will describe the technique of Josephson interferometry and review how it was implemented to verify that the high temperature cuprates had a dx2-y2 pairing symmetry.  I will then describe our ongoing quest to identify superconductors with complex order parameters that exhibit broken time-reversal symmetry, focusing on the ruthenate superconductor Sr2RuO4 which is suspected to exhibit a chiral p-wave order parameter of the form px±ipy, and on the heavy fermion superconductor UPt3 which exhibits two distinct superconductor phases.  In addition to exhibiting exciting new physics, complex superconductors have potential application for topologically-protected quantum computing.