Choquet boundaries in $K$-spaces

The article contains an account of the fundamental methods of Choquet theory. Decompositions, maximal operators, projectors, Choquet and Shilov boundaries are essential research tools in the fields of convex analysis, potential theory, approximation theory, geometry of convex surfaces, and so on.
The account is given in terms of the theory of Kantorovich spaces and the framework of a new and very general approach, which covers the majority of known constructions of the theory of integral representations.