  We discuss dense embeddings of surface groups and fully residually free
groups in topological groups. We show that a compact topological group contains
a nonabelian dense free group of finite rank if and only if it contains a dense
surface group. Also, we obtain a characterization of those Lie groups which
admit a dense faithfully embedded surface group. Similarly, we show that any
connected semisimple Lie group contains a dense copy of any fully residually
free group.
