  We show that bilayer graphene in the presence of a 2D superlattice potential
provides a highly tunable setup that can realize a variety of flat band
phenomena. We focus on two regimes: (i) topological flat bands with non-zero
Chern numbers, C, including bands with higher Chern numbers |C| > 1; and (ii)
an unprecedented phase consisting of a stack of nearly perfect flat bands with
C = 0. For realistic values of the potential and superlattice periodicity, this
stack can span nearly 100 meV, encompassing nearly all of the low-energy
spectrum. We further show that in the topological regime, the topological flat
band has a favorable band geometry for realizing a fractional Chern insulator
(FCI) and use exact diagonalization to show that the FCI is in fact the ground
state at 1/3 filling. Our results provide a realistic guide for future
experiments to realize a new platform for flat band phenomena.
