Modern theory of electrical networks: from the matrix-tree theorem to the theory of cluster varieties

The theory of electrical networks, in its current state, covers a number of areas of contemporary mathematics and mathematical physics including the combinatorics of paths, forests and groves on graphs, discrete harmonic analysis, problems of random walks, exactly solved models in statistical mechanics, cluster varieties related to spaces of totally positive matrices, discrete integrable systems, algebraic structures similar to Zamolodchikov's tetrahedron equation, and many others. The main aim of this survey is to present some of these topics, classical and recently discovered ones alike. Bibliography: 114 titles.