This tutorial will present recent efforts and results for surface acoustic wave (SAW) acoustoelectric amplification (AEA), which include modelling, analysis, experimental results and predictions. The topic of the acoustoelectric effect has been studied since the 1960s with a goal of producing a simple monolithic, continuous wave (CW), low power, low noise AEA. Although the conceptualization is understood, the practical implementation has been elusive. As material and fabrication technology has advanced, recent efforts have demonstrated embodiments and paths that may  achieve a practical AEA, having net gain, terminal gain, and CW operation. The tutorial will begin with a brief review of the wave propagation model as a foundation of past efforts of acoustoelectric theory and published experimental results into the 1980s. Next, a novel theoretical and modeling approach for a SAW AEA applicable to thin film overlay resistive or semiconductor material that is within or outside the acoustic path is developed. The theory is developed for a direct coupled SAW and thin film resistive interaction layer. The analysis uses the combination of coupling of modes and charge control analysis to develop a complete model describing the small signal operation and the large signal saturation effect. The model is developed for the case of direct coupling of thin films in the propagation path or coupling electrodes either inside or outside the propagation path. The approach describes the phenomenological physics and amplifier’s key parameters. Predictions are compared to published experimental results on lithium niobate using a monolayer graphene film and bonded film overlays that yielded continuous-wave operation and a net terminal gain. Experimental results that demonstrate the SAW amplification and attenuation properties will be discussed for communication and sensor applications.