  A general noncommutative-geometric theory of principal bundles is presented.
Quantum groups play the role of structure groups. General quantum spaces play
the role of base manifolds. A differential calculus on quantum principal
bundles is studied. In particular, algebras of horizontal and verticalized
differential forms on the bundle are introduced and investigated. The formalism
of connections is developed. Operators of horizontal projection, covariant
derivative and curvature are constructed and analyzed. A quantum generalization
of classical Weil's theory of characteristic classes is sketched. Quantum
analogs of infinitesimal gauge transformations are studied. Illustrative
examples and constructions are presented.

Quantum Algebra;Quantum Algebra