  In the paper we study properties of symmetric powers of complex manifolds. We
investigate a number of function theoretic properties (e. g. (quasi) $c$-finite
compactness, existence of peak functions) that are preserved by taking the
symmetric power. The case of symmetric products of planar domains is studied in
a more detailed way. In particular, a complete description of the
Carath\'eodory and Kobayashi hyperbolicity and Kobayashi completeness in that
class of domains is presented.
