Universal arrows to forgetful functors from categories of topological algebra

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and much more. An accent is put on relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and others may hopefully become amenable to methods of this theory.