The Mori Dream Space Conjecture asserts that the nef cone of the moduli space \overline M0,n of stable n-pointed rational curves is finitely generated and that every element is a semi-ample divisor. Conformal blocks divisors form full dimensional sub-cones of semi-ample divisors in the nef cone of \overline M0,n. In this talk I will discuss different ways to approach understanding whether or not these cones of conformal blocks divisors are finitely generated, and how that might relate to the Mori Dream Space Conjecture.