  We show that the recently observed superconductivity in twisted bilayer
graphene (TBG) can be explained as a consequence of the Kohn-Luttinger (KL)
instability which leads to an effective attraction between electrons with
originally repulsive interaction. Usually, the KL instability takes place at
extremely low energy scales, but in TBG, a doubling and subsequent strong
coupling of the van Hove singularities (vHS) in the electronic spectrum occurs
as the magic angle is approached, leading to extended saddle points in the
highest valence band (VB) with almost perfect nesting between states belonging
to different valleys. The highly anisotropic screening induces an effective
attraction in a $p$-wave channel with odd parity under the exchange of the two
disjoined patches of the Fermi line. We also predict the appearance of a
spin-density wave (SDW) instability, adjacent to the superconducting phase, and
the opening of a gap in the electronic spectrum from the condensation of spins
with wave vector corresponding to the nesting vector close to the vHS.
