  We prove very general formulae for the generating series of (Hodge) genera of
symmetric products with coefficients, which hold for complex quasi-projective
varieties with any kind of singularities, and which include many of the
classical results in the literature as special cases. Important specializations
of our results include generating series for extensions of Hodge numbers and
Hirzebruch genus to the singular setting and, in particular, generating series
for Intersection cohomology Hodge numbers and Goresky-MacPherson Intersection
cohomology signatures of symmetric products of complex projective varieties. A
very general proof is given based on Kuenneth formulae and pre-lambda
structures on the coefficient theory of a point. Moreover, Atiyah's approach to
power operations in K-theory also works in this context, giving a nice
description of the important related Adams operations. This last approach also
allows us to introduce very interesting coefficients on the symmetric products.
