  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.
