For a scheme X of finite type over a Noetherian scheme S we define a group of relative equidimensional cycles. We show that it is contravariantly functorial with respect to base change and thus provides a presheaf on the category of Noetherian schemes over S. Moreover this presheaf turns out to be a sheaf in a Grothendieck topology called the cdh-topology. The main goal of the paper is to study these Chow sheaves. In the particular case of X being a projective variety over a field of characteristic zero the Chow sheaf of effective cycles of dimension d is representable by the corresponding Chow variety. Even in this case though it turns out to be more convenient to work with sheaves than with varieties. In particular we construct certain short exact sequenecs of Chow sheaves which in the case of varieties over a field lead to localization long exact sequences in algebraic cycle homology and which do not have any obvious analog for Chow varieties.