  We examine both quantum and classical versions of the problem of spin
evolution in a slowly varying magnetic field. Main attention is given to the
first- and second-order adiabatic corrections in the case of in-plane
variations of the magnetic field. While the first-order correction relates to
the adiabatic Berry phase and Coriolis-type lateral deflection of the spin, the
second-order correction is shown to be responsible for the next-order geometric
phase and in-plain deflection. A comparison between different approaches,
including the exact (non-adiabatic) geometric phase, is presented.
