  The review is devoted to topological global aspects of quantal description.
The treatment concentrates on quantizations of kinematical observables ---
generalized positions and momenta. A broad class of quantum kinematics is
rigorously constructed for systems, the configuration space of which is either
a homogeneous space of a Lie group or a connected smooth finite-dimensional
manifold without boundary. The class also includes systems in an external gauge
field for an Abelian or a compact gauge group. Conditions for equivalence and
irreducibility of generalized quantum kinematics are investigated with the aim
of classification of possible quantizations. Complete classification theorems
are given in two special cases. It is attempted to motivate the global approach
based on a generalization of imprimitivity systems called {\em quantum Borel
kinematics}. These are classified by means of global invariants --- quantum
numbers of topological origin. Selected examples are presented which
demonstrate the richness of applications of Borel quantization. The review aims
to provide an introductory survey of the subject and to be sufficiently
selfcontained as well, so that it can serve as a standard reference concerning
Borel quantization for systems admitting localization on differentiable
manifolds.
