An atom exposed to an intense laser field may loose one or several of its electrons by photoionization. The field effectively turns all bound states into resonances of finite lifetime. For fields of constant intensity, this process is amenable to a time-independent description based on the method of complex scaling and on the Floquet theory of differential equations with periodic coefficients. The ionization rate of the state and the position of multiphoton resonances between Stark-shifted states can then be inferred from the spectrum of complex quasienergies of the Hamiltonian. After a general introduction to this approach, the talk will concentrate on a recent study of the origin of the sharp enhancements of emission of fast photoelectrons at certain intensities found a few years ago in the ionization of rare gases.