  In this paper we provide necessary and sufficient conditions for the
existence of the Kurzweil, McShane and Riemann product integrals of step
mappings with well-ordered steps, and for right regulated mappings with values
in Banach algebras. Our basic tools are the concepts of summability and
multipliability of families in normed algebras indexed by well-ordered subsets
of the real line. These concepts also lead to the generalization of some
results from the usual theory of infinite series and products. Finally, we
present two applications of product integrals: First, we describe the relation
between Stieltjes-type product integrals, Haahti products, and parallel
translation operators. Second, we provide a link between the theory of strong
Kurzweil product integrals and strong solutions of linear generalized
differential equations.
