Abstract. The Landau-Ginzburg Mirror Symmetry Conjecture states that for a quasi-homogeneous singularity W and a group G of symmetries of W , there is a dual singularity W T such that the orbifold A-model of W/G is isomorphic to the B-model of W T . The Landau-Ginzburg A-model is the Frobenius algebra H W,G constructed by Fan, Jarvis, and Ruan, and the B-model is the orbifold Milnor ring of W T . We verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group, and include a discussion of eight axioms which facilitate the computation of FJRW-rings.