  We study the influence of nonuniform magnetic fields on the magneto
conductance of mesoscopic microstructures. We show that the coupling of the
electron spin to the inhomogenous field gives rise to effects of the Berry
phase on ballistic quantum transport and discuss adiabaticity conditions
required to observe such effects. We present numerical results for different
ring geometries showing a splitting of Aharonov-Bohm conductance peaks for
single rings and corresponding signatures of the geometrical phase in weak
localization. The latter features can be qualitatively explained in a
semiclassical approach to quantum transport.
