For low-frequency, high-intensity laser fields, atomic and molecular targets can be ionized by multi-photon ionization (which is a nonlinear process and thus requires high intensity, but is nevertheless possible) as well as tunnel ionization (which can also happen in a sufficiently strong DC, or quasi-DC, field).
If you turn on the intensity further, above-threshold ionization can happen, in which the atomic potential barrier is pushed low enough that the ground state can escape. By this time the laser electric field comparable or stronger than the atomic one, which means that perturbation-theoretic concepts like photon absorption no longer mean much.
Apart from that, one very interesting phenomenon that can happen is the generation of very high order harmonics of the laser field. This is caused by the ionized electron being reaccelerated by the laser field towards the parent ion and colliding with it. The electron can then recombine and emit all of its considerable energy as a remarkably short (sub-femtosecond) pulse of light. This means that harmonics of order up to ~130 have been detected. A good reference for this is
M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, Anne L’Huillier, and P. B. Corkum. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49 no.3, pp. 2117–2132 (1994).