In this paper, we provide a broad overview of models and technologies pertaining to transcranial current brain stimulation (tCS), a family of related noninvasive techniques including direct current (tDCS), alternating current (tACS), and random noise current stimulation (tRNS). These techniques are based on the delivery of weak currents through the scalp (with electrode current intensity to area ratios of about 0.3-5 A/m2) at low frequencies (typically < 1 kHz) resulting in weak electric fields in the brain (with amplitudes of about 0.2-2 V/m). Here we review the biophysics and simulation of noninvasive, current-controlled generation of electric fields in the human brain and the models for the interaction of these electric fields with neurons, including a survey of in vitro and in vivo related studies. Finally, we outline directions for future fundamental and technological research.