Given a projective algebraic variety X, let Πp(X) denote the monoid of effective algebraic equivalence classes of effective algebraic cycles on X. The p-th Euler-Chow series of X is an element in the formal monoid-ring Z[[Πp(X)]] defined in terms of Euler characteristics of the Chow varieties Cp,α (X) of X, with α ∈ Πp(X). We provide a systematic treatment of such series, and give projective bu...