The study of electron transport in mesoscopic systems has recently turned to the observation of time dependent single electron e ects, where the electron transport is frequency locked to an external potential. Such devices are expected to form the basis of a standard of electric current, long sought after by the metrological community, to provide a representation of the ampere and to be compared with existing quantum standards of the volt and ohm. This thesis details new experimental investigations of one such system. The piezoelectric interaction between an acoustic wave travelling on the surface of a GaAs heterostructure and electrons in a quasi-1D sys- tem de ned therein generates a current which, under certain conditions, can be quantized in units of $e\; f$ where $e$ is the electron charge and $f$ the surface acoustic wave frequency. The general conditions under which this $single\; electron\; acoustoelectric\; effect$ is observable are studied, and experimental results presented which demonstrate that the e ffect represents a possible route towards a current standard. The precision of the e ffect is assessed in a variety of experimental situations and device geometries. Several ways to enhance the precision of the eff ect are presented. Firstly a weak counterpropagating SAW beam produces a dynamic tuning of the SAW potential. Observations of a quantized acoustoelectric current are then presented in novel etched-channel SAW devices which aff ord a more precise current by allowing better control over the channel geometry. The presently attainable precision of the technique is at the level of 10's of ppm. Detailed measurements are presented of the single electron acoustoelectric e ffect in a magnetic fi eld applied perpendicular to the two-dimensional electron gas. Commensurability oscillations are observed for the interval of current between acoustoelectric current plateaux when the cyclotron diameter and SAW wavelength are comparable. The oscillations show a particular phase dependence which results in an oscillating plateau slope as a function of applied magnetic fi eld. Results are also presented from measurements of the interaction between a surface acoustic wave and open 1D systems. Here the quantized current is not observed, but instead the behaviour of the measured current depends sensitively on the geometry of the channel. Two situations are possible in this regime. Interaction between the SAW and slow electrons in the uppermost 1D subband within the channel produces an oscillatory acoustoelectric current as a function of subband occupancy. These oscillations are observed in all subbands of clean constrictions for the first time. Secondly, interaction between the SAW and electrons in the device leads causes a contribution to the acoustoelectric current which is proportional to the quantized channel conductance, this contribution dominating transport in certain device geometries.