Boyer, Lance L.Since 2006 it has been discovered experimentally that the superconducting state spontaneously breaks time-reversal symmetry (TRS) in several materials, such as Sr2RuO4, UPt3, URu2Si2, PrOs4Sb12, and Bi/Ni bilayers. This dissertation studies three physical phenomena related to time-reversal symmetry breaking (TRSB) in these superconductors. The experimental evidence for TRSB comes from the magneto-optical polar Kerr effect, which is determined by the high frequency ac Hall conductivity. However, these superconductors are also expected to exhibit a spontaneous dc Hall effect in the absence of an applied magnetic field. In the first part of this dissertation we propose a method for measuring the low frequency Hall conductivity in superconductors with TRSB. The method is based on a Corbino disk geometry where an oscillating co-axial magnetic field induces circular electric field, which, in turn, induces radial charge oscillations due to the Hall conductivity. In the second part, we propose an explanation for the polar Kerr effect observed in the Hidden-Order phase of the heavy-fermion superconductor URu2Si2. Using a Ginzburg-Landau model for a complex order parameter, we show that the system can have a metastable ferromagnetic state, which produces the Kerr signal, even if the Hidden-Order state respects TRS. We predict that applying a reversed magnetic field should reset the system to the non-magnetic ground state, resulting in zero Kerr signal. In the third part of the dissertation, we investigate the conditions for the existence of a Majorana bound state on a vortex in a 2D d+id superconductor with strong spin-orbit coupling. This TRSB pairing was proposed earlier for the Ni/Bi bilayer. We find that the Majorana bound state can exist for a d+id pairing under conditions similar to those for s-wave pairing.enDissertationCondensed matter physicsHall EffectHidden OrderMajoranaSuperconductivitySuperconductorTime-Reversal Symmetry