  The famous Banach-Mazur problem, which asks if every infinite-dimensional
Banach space has an infinite-dimensional separable quotient Banach space, has
remained unsolved for 85 years, though it has been answered in the affirmative
for reflexive Banach spaces and even Banach spaces which are duals. The
analogous problem for locally convex spaces has been answered in the negative,
but has been shown to be true for large classes of locally convex spaces
including all non-normable Fr\'echet spaces. In this paper the analogous
problem for topological groups is investigated. Indeed there are four natural
analogues: Does every non-totally disconnected topological group have a
separable quotient group which is (i) non-trivial; (ii) infinite; (iii)
metrizable; (iv) infinite metrizable. All four questions are answered here in
the negative. However, positive answers are proved for important classes of
topological groups including (a) all compact groups; (b) all locally compact
abelian groups; (c) all $\sigma$-compact locally compact groups; (d) all
abelian pro-Lie groups; (e) all $\sigma$-compact pro-Lie groups; (f) all
pseudocompact groups. Negative answers are proved for precompact groups.
