In this talk, I will discuss generalizations of rationally connected fibrations and the MRC quotient.  In particular, we will define a fibration as Chow constant if pushforward induces an isomorphism on the Chow group of 0-cycles and as cohomologically constant if pullback induces an isomorphism on p-forms.  We will show that maximal Chow constant and cohomologically constant fibrations exist and also construct maximal Chow trivial and cohomologically trivial fibrations.  We will discuss applications, consequences, and the relationship to the generalized Bloch conjecture.  This is joint work with David Stapleton.