In this talk, we will cover some results on the nature of mixed Hodge structures on the cohomology of complex quasi-projective varieties. Specifically, we will be interested in varieties whose cohomology behaves as an exterior algebra. For those, we will manage to deduce the mixed Hodge structure on their symmetric products. These results will allows us to obtain a general formula for the mixed Hodge structure of free abelian character varieties for some reductive groups. We will study this formula in some detail for G=Sp(n,C), a case handled by me and C. Florentino by adapting our results for G=GL(n,C) and SL(n,C). Another interesting class of varieties whose cohomology is an exterior algebra is given by connected complex linear algebraic groups. For those, we will calculate the mixed Hodge structure on their symmetric products, and so obtain some facts on their motivic Zeta function — a very good example of what is mentioned by Ravi Vakil as a “baby motif”.