In [1] and [2]Baum, Fulton and MacPherson, generalizing the celebrated Grothendieck-Riemann-Roch theorem, proved that given a category 1/ of quasi-projective schemes there is a natural transformation called the Todd class of functors (covariant for proper morphisms) between K' 0 , the homology algebraic ^-theory of coherent sheaves and any of the standard homology theories. Here we announce generalizations of the results of [1] and [2] to Quillen's higher algebraic ^-theory [8] which may help to illuminate the relationship between algebraic AT-theory and more ordinary cohomology theories.The statements of our theorems depend on defining global analogues of Quillen's construction of Chern classes for the ^-theory of a ring [3], [9]. We can use any of the standard cohomology theories defined on 1/, such as étale or crystalline cohomology or even the Chow ring. All of these theories can be realized for each X G 1/ as the hypercohomology of a graded complex or procomplex T* 9 j E Z, of sheaves on the Zariski site of X. All of these theories have Chern classes for representations of sheaves of groups and there exist universal classes Q G H di (X, GL(0 X whose domains are the relative tf-groups, defined so as to force a Quillen-style localization sequence. One can show that these classes coincide for p = 0 with those of Iversen [7]. For p > 0 they are group homomorphisms and are compatible with products in the way described by Bloch [3], hence one can define a Chern character with supports, which is a ring homomorphism chï : © KJX, X-Y) -* © H^(Z, rf ) ® Q.