A generalization of the classical gauge theory is presented, in which compact quantum groups play the role of the internal symmetry groups. All considerations are performed in the framework of a noncommutative-geometric formalism of locally trivial quantum principal bundles over classical smooth manifolds. Quantum counterparts of classical gauge bundles, and classical gauge transformations, are introduced and investigated. A natural differential calculus on quantum gauge bundles is constructed and analysed. Kinematical and dynamical properties of corresponding gauge theories are discussed. Particular attention is given to the purely quantum phenomena appearing in the formalism, and their physical interpretation. An example with quantum SU(2) group is considered.