The authors develop a generalized notion of principal bundles and connections. This notion includes a previous generalization as a special case. The paper is very algebraic and fairly technical. The role of the gauge group is played by a coalgebra, and the role of a principal bundle is played by an algebra. The most explicit example given in this paper is the quantum cylinder, i.e., a free associative algebra generated by \(x\), \(x^{-1}\) and \(y\) with the relations \(yx=qxy\), \(xx^{-1}=x^{-1}=x^{-1}x=1\) with a natural Hopf algebra structure. This is an elegant collection of definitions. A nice explanation of the background material needed for this paper is contained in [the authors, Commun. Math. Phys. 157, No. 3, 591-638 (1993; Zbl 0817.58003)], that paper also explains how an ordinary principal bundle generates a generalized principal bundle.