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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 isrigorously constructed for systems, the configuration space of which is eithera homogeneous space of a Lie group or a connected smooth finite-dimensionalmanifold without boundary. The class also includes systems in an external gaugefield for an Abelian or a compact gauge group. Conditions for equivalence andirreducibility of generalized quantum kinematics are investigated with the aimof classification of possible quantizations. Complete classification theoremsare given in two special cases. It is attempted to motivate the global approachbased on a generalization of imprimitivity systems called {\em quantum Borelkinematics}. These are classified by means of global invariants --- quantumnumbers of topological origin. Selected examples are presented whichdemonstrate the richness of applications of Borel quantization. The review aimsto provide an introductory survey of the subject and to be sufficientlyselfcontained as well, so that it can serve as a standard reference concerningBorel quantization for systems admitting localization on differentiablemanifolds.Comment: 43 page

QUANTIZATION OF KINEMATICS ON CONFIGURATION MANIFOLDS