We study two kinds of quantum interference effects in transportthe Aharonov-Bohm effect and the weak-localization effectin quasi-one-dimensional wires and rings to address issues concerning the phase-coherence length, spin-orbit scattering, and the flux cancellation mechanism which is predicted to be present when the elastic mean free path exceeds the sample width. Our devices are fabricated on GaAs/AlxGa1-xAs and pseudomorphic GaxIn1-xAs/AlxIn1- xAs heterostructure materials and the experiments carried out at 0.420 K temperatures. In the GaAs/AlxGa1-xAs devices which exhibit no significant spin-orbit scattering, we were able to extract a phase-coherence length l from the amplitude of the Aharonov-Bohm magnetoresistance oscillations in different sized rings. We find it to be in agreement with l deduced from the weak-localization data in parallel wires when the one-dimensional weak-localization theory including the flux cancellation mechanism is used to fit the data. We therefore unambiguously establish that the same l governs the behavior of the two quantum interference phenomena of Aharonov-Bohm oscillations and weak localization, and that the flux cancellation is in force. In the pseudomorphic GaxIn1-xAs/AlxIn1-xAs heterostructure devices which exhibit strong spin-orbit interaction effects, l exceeds the spin-orbit-scattering length at low temperatures. The amplitude of Aharonov-Bohm oscillations can only be explained by introducing reduction factors due to spin-orbit scattering.