In noncommutative geometry there are several versions of noncommutative bundle theory, e.g. considering vector bundles as finitely generated projective modules and the theory of noncommutative principal bundles as Hopf-Galois extensions and their coalgebra and global analogues. Each of these formalism can be a setup in whcih one can try to develop a noncommutative analogue of the theory of connections on a bundle. The connection theory on noncommutative spaces is of course, the basis of gauge theories on noncommutative spaces. A remarkable distinction between the commutative and noncommutative case of connections on Hopf-Galois extensions is the difference between generic connection on a generic and so-called strong connections, as discovered by P. Hajac. There is an approach close to Koszul’s for dga-s, by defining the connections by action on noncommutative differential forms of Karoubi or by an appropriate analogue in cyclic homology.
There is a rather vast literature on the subject and we should list the more important ones.
Max Karoubi, Connexions, courbures et classes caractéristiques en K-théorie algébrique, Current trends in algebraic topology, Part I, vol. 2, 19-27, London, Ontario 1981, pdf
Alain Connes, Noncommutative geometry, Academic Press 1994, 661 p. PDF