Instantons and monopoles

In this survey we present the main notions and constructions of gauge theories, namely, the Donaldson theory, the Seiberg–Witten theory, and the theory of B-monopoles, which connects the previous two theories. In the framework of differential geometry these theories give new invariants of smooth structures in dimension 4. The introduction of these new gauge invariants has helped to solve many problems of modern geometry. The apparatus developed in the framework of these theories leads to new modern methods of investigation both in smooth geometry and in applied problems of mathematical physics. Without striving for the greatest possible generality, the survey aims to present the topic in maximal breadth and accessibility.