  We survey and expand on the work of Segal, Milgram and the author on the
topology of spaces of maps of positive genus curves into $n$-th complex
projective space, $n\geq 1$ (in both the holomorphic and continuous
categories). Both based and unbased maps are studied and in particular we
compute the fundamental groups of the spaces in question. The relevant case
when $n=1$ is given by a non-trivial extension which we fully determine.
