  Let $X$ be a smooth projective real algebraic variety. We give new positive
and negative results on the problem of approximating a submanifold of the real
locus of $X$ by real loci of subvarieties of $X$, as well as on the problem of
determining the subgroups of the Chow groups of $X$ generated by subvarieties
with nonsingular real loci, or with empty real loci.
