In this article, we show how the classical theory of principal fibre bundles (in short PFB) transcribes into a quantum group formalism. In this dual formulation, a PFB is given by a right comodule algebra over a Hopf algebra with a mapping satisfying certain compatibility properties. In our case, is the (commutative) -algebra of complex-valued continuous functions on the total space P and is the Hopf algebra of complex-valued functions on the structure group G. These underlying spaces P and G are endowed with a topology only. The subalgebra of -invariant elements is identified with the algebra of complex-valued functions on the base space B. In order to define horizontal one-forms, a differential calculus is needed. Since no a priori differential structure is assumed, we use the calculus of the universal differential envelope which can be defined on any unital algebra. A connection on the PFB is then defined by a splitting of the universal one-forms as a direct sum of horizontal and vertical subspaces: In case of a strong connection in a trivial PFB, the general expressions of the connection one-form and the curvature two-form are given. Following [5] and [6], a locally trivial PFB can be constructed through a gluing procedure of a cover of the algebra We conclude quoting two applications of our approach and with an Appendix containing some basic notions of Hopf algebras theory. In [1] and [9], we find readable introductions, for a thorough treatment, see [7] and the references therein.