  A remarkable recent experiment has observed Mott insulator and proximate
superconductor phases in twisted bilayer graphene when electrons partly fill a
nearly flat mini-band that arises a `magic' twist angle. However, the nature of
the Mott insulator, origin of superconductivity and an effective low energy
model remain to be determined. We propose a Mott insulator with intervalley
coherence that spontaneously breaks U(1) valley symmetry, and describe a
mechanism that selects this order over the competing magnetically ordered
states favored by the Hunds coupling. We also identify symmetry related
features of the nearly flat band that are key to understanding the strong
correlation physics and constrain any tight binding description. First,
although the charge density is concentrated on the triangular lattice sites of
the moir$\text{\'e }$ pattern, the Wannier states of the tight-binding model
must be centered on different sites which form a honeycomb lattice. Next,
spatially localizing electrons derived from the nearly flat band necessarily
breaks valley and other symmetries within any mean-field treatment, which is
suggestive of a valley-ordered Mott state, and also dictates that additional
symmetry breaking is present to remove symmetry-enforced band contacts.
Tight-binding models describing the nearly flat mini-band are derived, which
highlight the importance of further neighbor hopping and interactions. We
discuss consequences of this picture for superconducting states obtained on
doping the valley ordered Mott insulator. We show how important features of the
experimental phenomenology may be explained and suggest a number of further
experiments for the future. We also describe a model for correlated states in
trilayer graphene heterostructures and contrast it with the bilayer case.
