Let G be a complex affine algebraic group. If C is a smooth algebraic curve and x is a point in C, the affine Grassmannian is an algebro-geometric object that parametrizes G-bundles on C together with a trivialization outside x. Alternatively, one can define the affine Grassmannian as the quotient G((t))/G[[t]]. In this talk, we present possible analogues for the affine Grassmannian, in the setting where the curve is replaced by a smooth projective surface, and the trivialization data are specified with respect to flags of closed subschemes. We also obtain parallel descriptions in terms of quotients of the double loop group G((t))((s)). Based on a joint work A. Maffei and G. Vezzosi.