  Consider a complex abelian variety X on which a field F acts. Generalizing a
construction of A. Weil, one associates to this a subspace W_F of the
cohomology of X, which we call the space of Weil classes w.r.t. F. The purpose
of this paper is to answer the following two questions: Q1: under what
conditions on F does the space W_F contain, or even consist of, Hodge classes?,
Q2: if W_F contains Hodge classes, under what conditions on F are these
exceptional? In case X is defined over a number field, we also answer the
analogous questions for Tate classes.
