  We present a systematic study on the influence of Rashba spin-orbit coupling,
interlayer potential difference and exchange field on the topological
properties of bilayer graphene. In the presence of only Rashba spin-orbit
coupling and interlayer potential difference, the band gap opening due to
broken out-of-plane inversion symmetry offers new possibilities of realizing
tunable topological phase transitions by varying an external gate voltage. We
find a two-dimensional $Z_2$ topological insulator phase and a quantum valley
Hall phase in $AB$-stacked bilayer graphene and obtain their effective
low-energy Hamiltonians near the Dirac points. For $AA$ stacking, we do not
find any topological insulator phase in the presence of large Rashba spin-orbit
coupling. When the exchange field is also turned on, the bilayer system
exhibits a rich variety of topological phases including a quantum anomalous
Hall phase, and we obtain the phase diagram as a function of the Rashba
spin-orbit coupling, interlayer potential difference, and exchange field.
