  A theory of principal bundles possessing quantum structure groups and
classical base manifolds is presented. Structural analysis of such quantum
principal bundles is performed. A differential calculus is constructed,
combining differential forms on the base manifold with an appropriate
differential calculus on the structure quantum group. Relations between the
calculus on the group and the calculus on the bundle are investigated. A
concept of (pseudo)tensoriality is formulated. The formalism of connections is
developed. In particular, operators of horizontal projection, covariant
derivative and curvature are constructed and analyzed. Generalizations of the
first structure equation and of the Bianchi identity are found. Illustrative
examples are presented.
