As we know, Grothendieck Riemann Roch only involves $K_{0}$. Is there any work generalizing this formula to (Quillen's Higher K)? If there is, what is the meaning for such kind of formula?
Thanks in advance
As we know, Grothendieck Riemann Roch only involves $K_{0}$. Is there any work generalizing this formula to (Quillen's Higher K)? If there is, what is the meaning for such kind of formula? Thanks in advance 


MR0624666 (83m:14013) Gillet, Henri RiemannRoch theorems for higher algebraic Ktheory. Adv. in Math. 40 (1981), no. 3, 203–289. 


There is the recent paper Algebraic Ktheory, $A^1$homotopy and RiemannRoch theorems by Riou, which gives a different proof of the results by Gillet referenced in the answer by profilesdroxford54. Abstract:
To give you a rough idea, algebraic Ktheory is representable in the stable homotopy category arising from $A^1$homotopy theory of schemes over a regular scheme. As far as I understand it, GrothendieckRiemannRoch is just a shadow of the maps from the spectrum representing algebraic Ktheory to the EilenbergMacLane spectrum. But the paper above considers more than just stable homotopy. 

