  Twisted trilayer graphene (TLG) may be the simplest realistic system so far,
which has flat bands with nontrivial topology. Here, we give a detailed
calculation about its band structures and the band topology, i.e. valley Chern
number of the nearly flat bands, with the continuum model. With realistic
parameters, the magic angle of twisted TLG is about 1.12$^{\circ}$, at which
two nearly flat bands appears. Unlike the twisted bilayer graphene, a small
twist angle can induce a tiny gap at all the Dirac points, which can be
enlarged further by a perpendicular electric field. The valley Chern numbers of
the two nearly flat bands in the twisted TLG depends on the twist angle
$\theta$ and the perpendicular electric field $E_\perp$. Considering its
topological flat bands, the twisted TLG should be an ideal experimental
platform to study the strongly correlated physics in topologically nontrivial
flat band systems. And, due to its reduced symmetry, the correlated states in
twisted TLG should be quite different from that in twisted bilayer graphene and
twisted double bilayer graphene.
