We report on the effects of laser illumination on the electronic properties of bilayer graphene. By using Floquet theory combined with Green's functions, we unveil the appearance of laser-induced gaps not only at integer multiples of h Omega/2 but also at the Dirac point with features which are shown to depend strongly on the laser polarization. Trigonal warping corrections are shown to lead to important corrections for radiation in the terahertz range, reducing the size of the dynamical gaps. Furthermore, our analysis of the topological properties at low energies reveals that, when irradiated with linearly polarized light, ideal bilayer graphene behaves as a trivial insulator, whereas circular polarization leads to a nontrivial insulator per valley.