The Chow and Hilbert quotients of a projective variety X by a torus T are nice canonical quotients. In the case where X has a degeneration to a toric variety, the Chow and Hilbert quotients of X also degenerate to toric varieties. An important example of this is the moduli space overline{M}_{0,n} which inherits toric degenerations from the degenerations of the Grassmannian G(2,n). I will recall the details of the Chow and Hilbert quotients and (Grobner) degenerations before describing joint work with Angela Gibney giving an explicit description of these toric degenerations.