  We provide a general method for computing rational Chow rings of moduli of
smooth complete intersections. We specialize this result in different ways: to
compute the integral Picard group of the associated stack ; to obtain an
explicit presentation of rational Chow rings of moduli of smooth complete
intersections of codimension two; to prove old and new results on moduli of
smooth curves of genus $\leq 5$ and polarized K3 surfaces of degree $\leq 8$.
