Abstract: Back in the 1930's, Pauling came up with an idea of approximating electron wave functions in complicated molecules by functions on a graph: the atoms are vertexes of this graph, and bonds are the edges. A wave function is an eigenfunction on the Schroedinger operator on the graph, which is a one-dimensional variety, rather than a combinatorial graph. Such a structure got a name of a quantum graph. Recent advances in nano-technology drew attention to the study of systems in narrow tubes, and such system can also be approximated by quantum graphs. In the talk, I will discuss both the approximation problems and the properties of quantum graphs.