  Twisted bilayer graphene displays insulating and superconducting phases
caused by exceptional flattening of its lowest energy bands. Superconductivity
with highest $T_c$ appears at hole and electron dopings, near half-filling for
valence or conduction bands. In the hole-doped case, the data show that
three-fold lattice rotation symmetry is broken in the superconducting phase,
i.e., a superconductor is also a nematic. We present the mechanism for nematic
superconductivity. We take as an input the fact that at relevant dopings the
Fermi energy lies in the vicinity of twist-induced Van Hove singularities in
the density of states and argue that the low-energy physics can be properly
described by patch models with six Van Hove points for electron doping and
twelve Van Hove points for hole doping. We obtain pairing interactions for the
patch models in terms of parameters of the microscopic model for the flat
bands, which contains both local and twist-induced non-local interactions and
show that the latter gives rise to attraction in different superconducting
channels. For hole-doping, we find two attractive channels, $g$ and $i$-waves,
with almost equal coupling constants. We show that in the co-existence state,
where both order parameters are non-zero, the three-fold lattice rotation
symmetry is broken, i.e., a superconductor is also a nematic. We find two
possible nematic states, one is time-reversal symmetric, the other additionally
breaks time-reversal symmetry. Our scenario for nematic superconductivity is
based on generic symmetry considerations, and we expect it to be applicable
also to other systems with two (or more) attractive channels with similar
couplings.
