The influence of periodic driving on coherent tunneling is investigated, using a quartic double-well potential in a monochromatic external field as a working example. Extensive numerical studies of the long-time behaviour of this system are combined with an analytical description on basis of the quasienergy formalism. Approximate solutions of the dynamics are possible in the two opposite limits of adiabatic and fast driving, respectively. In both cases, the tunneling rate is enhanced, compared with the unperturbed value. This is confirmed by our numerical results. For driving frequencies in the range of the bare tunnel splitting novel types of tunneling behaviour occur, including localization of the wave packet in one of the wells (coherent destruction of tunneling). They can be explained in terms of the local Floquet spectrum and are verified by the time evolution of quantum probabilities.