  The theory of general Galois-type extensions is presented, including the
interrelations between coalgebra extensions and algebra (co)extensions,
properties of corresponding (co)translation maps, and rudiments of entwinings
and factorisations. To achieve broad perspective, this theory is placed in the
context of far reaching generalisations of the Galois condition to the setting
of corings. At the same time, to bring together K-theory and general Galois
theory, the equivariant projectivity of extensions is assumed resulting in the
centrepiece concept of a principal extension. Motivated by noncommutative
geometry, we employ such extensions as replacements of principal bundles. This
brings about the notion of a strong connection and yields finitely generated
projective associated modules, which play the role of noncommutative vector
bundles. Subsequently, the theory of strong connections is developed. It is
purported as a basic ingredient in the construction of the Chern character for
Galois-type extensions (called the Chern-Galois character).
