  We propose that the electronic structure of twisted bilayer graphene (TBG)
can be understood as Dirac fermions coupled with opposite pseudo magnetic
fields generated by the moir\'e pattern. The two low-energy flat bands from
each monolayer valley originate from the two zeroth pseudo Landau levels of
Dirac fermions under such opposite effective magnetic fields, which have
opposite sublattice polarizations and carry opposite Chern numbers $\pm1$,
giving rise to helical edge states in the gaps below and above the low-energy
bulk bands near the first magic angle. We argue that small Coulomb interactions
would split the eight-fold degeneracy (including valley and physical spin) of
these zeroth pseudo Landau levels, and may lead to insulating phases with
non-vanishing Chern numbers at integer fillings. Besides, we show that all the
high-energy bands below or above the flat bands are also topologically
nontrivial in the sense that for each valley the sum of their Berry phases is
quantized as $\pm\pi$. Such quantized Berry phases give rise to nearly flat
edge states, which are dependent on truncations on the moir\'e length scale.
Our work provides a complete and clear picture for the electronic structure and
topological properties of TBG, and has significant implications on the natrue
of the correlated insulating phase observed in experiments.
