  Geometric phases are a universal concept that underpins numerous phenomena
involving multi-component wave fields. These polarization-dependent phases are
inherent in interference effects, spin-orbit interaction phenomena, and
topological properties of vector wave fields. Geometric phases have been
thoroughly studied in two-component fields, such as two-level quantum systems
or paraxial optical waves. However, their description for fields with three or
more components, such as generic nonparaxial optical fields routinely used in
modern nano-optics, constitutes a nontrivial problem. Here we describe
geometric, dynamical, and total phases calculated along a closed spatial
contour in a multi-component complex field, with particular emphasis on 2D
(paraxial) and 3D (nonparaxial) optical fields. We present several equivalent
approaches: (i) an algebraic formalism, universal for any multi-component
field; (ii) a dynamical approach using the Coriolis coupling between the spin
angular momentum and reference-frame rotations; and (iii) a geometric
representation, which unifies the Pancharatnam-Berry phase for the 2D
polarization on the Poincar\'e sphere and the Majorana-sphere representation
for the 3D polarized fields. Most importantly, we reveal close connections
between geometric phases, angular-momentum properties of the field, and
topological properties of polarization singularities in 2D and 3D fields, such
as C-points and polarization M\"obius strips.
