What is the behavior of a low-dimensional quantal system whose classical, deterministic, Hamiltonian, counterpart is non-integrable and undergoes a transition to chaos? After a general introduction this report engages the question by focussing on hydrogen atoms prepared with principal quantum number n0 1 being driven by a linearly polarized, periodic electric field strong enough to cause "ionization"; this means true ionization plus excitation to above an experimentally determined n-cutoffnc > n0. This is one of the few time-dependent systems on which experiments, quantal calculations, and classical calculations have been done in sufficient depth to show that answers to the question range from simple to subtle. The dynamical behavior of the system changes with increasing scaled frequency, which classically is the ratio ¿/¿K of the driving frequency ¿ and the unperturbed Kepler frequency ¿K. Quantally, this corresponds to n03¿ in atomic units. Comparisons among experimental data and quantal and classical theoretical calculations have so far revealed six different regimes of dynamical behavior for different ranges of n03¿. After describing all six, this report emphasizes the first three, or n03¿ up to about 1.2. Described in detail are experiments carried out at Stony Brook with n0 = 32,…, 90 hydrogen atoms being driven by an ¿/2p = 9.92 GHz field, or n03¿ = 0.05-1.1 (subsequently extended down to n0 = 24, or n03¿ = 0.021). The data show the quantal system being influenced by various resonance effects, some of whose origins are most easily found in the corresponding classical system, others of which are not. When ¿/¿K is near low-order rational fractions r/s, R = 1,2 and S = 1,2,…, the classical dynamics is strongly affected by nonlinear resonances easily visualized in computed stroboscopic phase portraits of the 1d motion. The trapping of orbits inside them leads to classical local stability. Where the quantitative agreement between experimental data and classical calculations is good for threshold field amplitudes for the onset of "ionization", the classical theory gives keen insight into the semiclassical dynamics. Conversely, where the quantitative agreement breaks down is a signature for the importance of quantal effects. Often this occurs where the nonclassical behavior is, nevertheless, still anchored in subtle ways to the classical dynamics in and near nonlinear resonances. The report includes a detailed, critical comparison among data sets for n03¿ = 1.1 obtained from experiments in different laboratories, using either excited hydrogen atoms or alkali Rydberg atoms prepared in hydrogen-like states with small quantum defects. It also includes a careful discussion of experimental data obtained with a static electric field superimposed with the microwave electric field. The data demonstrate that the static electric field may be used to fine-tune the scaled frequency, which is likely to be exploited to advantage in future experiments.