Graphene is the most two-dimensional platform currently available
as a host for an electron gas, and offers promise to make observable a variety of
effects in a perpendicular magnetic field. For example, recent advances in aligning
graphene on a boron-nitride substrate have led to the creation of high-quality
Moire patterns with large unit cells. Similar large-cell superlattices can also be
created in twisted bilayers. In a perpendicular magnetic field, near zero energy
the periodicity has little effect on the spectrum, but with increasing energy the
spectrum evolves into the much-anticipated Hofstadter butterfly. The crossover
between these behaviors is controlled by a saddle point in the zero-field spectrum.
We demonstrate through a semiclassical analysis how the quantization of orbits changes
as the saddle point is crossed, allowing the richness of the Hofstadter spectrum to emerge
above it, and discuss some possible experimental consequences. We then consider
graphene systems in much higher magnetic fields -- the quantum Hall regime -- where
transport is controlled by edge states. For undoped graphene these edge states
may have a helical nature. We discuss what happens when an "internal edge" is
created in bilayer graphene using a split gate geometry, where a surprisingly rich
internal structure emerges with a number of possible states. Such a geometry admits
transport probes which potentially reveal different aspects of the internal structure, and we
discuss our expectations for how the state of this internal edge can be reflected in
such measurements.