On solvability and unsolvability of equations in explicit form

In this survey the classical results of Abel, Liouville, Galois, Picard, Vessiot, Kolchin, and others on the solvability and unsolvability of equations in explicit form are discussed. The one-dimensional topological version of Galois theory is presented in detail (this version describes topological obstructions to the representability of functions by quadratures).