The emergence of vehicle automation and its subsequent growth has led to new transport service offerings, generally known as Autonomous Mobility Services (AMS), that have the potential to replace human-operated vehicles. However, the functionalities of AMS are increasingly blurring the fine lines that currently distinguish different transport modes. For example, an autonomous shuttle bus, a form of autonomous transit, may serve a similar function as an autonomous taxi/robo-taxi, both coinciding with the concept of Shared Autonomous Mobility Services (SAMS). Even if the functionalities or operational principles are different, people may perceive sharing rides in any of these services as alike. Similarly, the absence of a human driver makes the concepts of autonomous carsharing and ridehailing similar. Hence, there is a need to review studies related to SAMS. However, few studies have attempted to perform a comprehensive review of public acceptance of SAMS. This study aims to fill this gap by reviewing studies related to public acceptability and acceptance, perception, intention to use, attitudes, mode choice and willingness constructs regarding SAMS. This review clearly distinguishes different types of SAMS while examining public’s acceptability and acceptance of SAMS across five dimensions: perception about the services, intention to choose and use those over other modal alternatives, frequency of usage and willingness constructs. Overall, the results from our review indicate the presence of heterogeneity across sub-groups regarding the adoption of SAMS. Discussing the factors affecting SAMS acceptance in a detailed manner, our study serves to provide a stocktake of the progress in this genre of research.