Automated mobility has always conveyed fantasies about its ability to meet future mobility needs and challenges. If research is still debating the when, where and how of this mobility disruption, there seems to be a consensus on its advent. Meanwhile, the investigation of the demand for this mobility is of critical importance. Stated preference surveys are a common and powerful tool to foresee this demand. This paper tackles the question of the acceptance of shared autonomous vehicles (SAVs) by reviewing most significant published papers on travel demand and travel surveys for SAVs. Given the recent developments in the last three years, the paper updates and complements previous literature reviews. In contrast with previous research, this paper shows that the impact of various factors on the intention to use SAVs is still controversial. This includes age, gender, income or car ownership. We identify most consensual and controversial effects and correspondingly suggest future research tracks to address some the identified gaps.