  The moir\'e superlattice of misaligned atomic bilayers paves the way for
designing a new class of materials with wide tunability. In this work, we
propose a photonic analog of the moir\'e superlattice based on dielectric
resonator quasi-atoms. In sharp contrast to van der Waals materials with weak
interlayer coupling, we realize the strong coupling regime in a moir\'e
superlattice, characterized by cascades of robust flat bands at large
twist-angles. Surprisingly, we find that these flat bands are characterized by
a non-trivial band topology, the origin of which is the moir\'e pattern of the
resonator arrangement. The physical manifestation of the flat band topology is
a robust one-dimensional conducting channel on edge, protected by the
reflection symmetry of the moir\'e superlattice. By explicitly breaking the
underlying reflection symmetry on the boundary terminations, we show that the
first-order topological edge modes naturally deform into higher-order
topological corner modes. Our work pioneers the physics of topological phases
in the designable platform of photonic moir\'e superlattices beyond the weakly
coupled regime.
