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We discuss spin-dependent quantum transport experiments performed in narrow-gap semiconductors and in semimetals with substantial spin-orbit interaction (SOI). Samples consist of InSb, InAs, InGaAs or Bi, patterned into mesoscopic geometries. SOI modifies weak-localization to weak anti-localization (WAL), which is sensitive to spin- and phase coherence lengths. Using WAL we measure the dependence of spin coherence lengths on the wire width in narrow nanolithographic ballistic InSb wires, ballistic InAs wires, and diffusive Bi wires. In all three systems we find that the spin coherence lengths increase with decreasing wire width. The measurements also indicate that Bi has surface states with strong Rashba-like SOI. We further used WAL to study the magnetic interactions between a 2D electron accumulation layer at the InAs surface and local magnetic moments from rare earth and transition metal ions. In InAs heterostructure mesoscopic ring geometries, we observe coexisting Aharonov-Bohm oscillations due to spatial quantum interference, and Altshuler-Aronov-Spivak oscillations due to time-reversed paths. A quantum coherent network formalism shows that the quantum phase coherence length coincides for both spatial- and temporal-loop quantum interference, indicating a fundamentally similar origin. We also describe our search for edge states from confinement-induced SOI in InGaAs heterostructures, which uses the duality between the Aharonov-Bohm and the Aharonov-Casher phase. And finally we present data on an unusual interface magnetoresistance in InSb/CoFe geometries.