We study classical and quantum scattering in periodically driven mesoscopic electronic devices. The ac-driven square potential well, a prototypical example, is shown to be a chaotic scatterer. Quantum transport in the nonlinear regime of strong driving is investigated in the Floquet framework. In the deep quantum regime, scattering into and out of bound states of an effective potential explains most of the structure in reflection, transmission, and dwell time, controllable via the external driving. For large amplitude and frequency, a single ac-driven quantum well exhibits dynamical tunneling. We identify signatures of the classical phase-space structure in Wigner-function representations of propagated wavepackets.