Assertions are one of the most useful automated techniques for checking program's behaviour and hence have been used for different verification and validation tasks. We provide an overview of the last two decades of research involving "assertions" in software testing. Based on a term-based search, we filtered the inclusion of relevant papers and synthesised them with respect to the problem addressed, the solution designed, and the evaluation conducted. The survey rendered 145 papers on assertions in software testing. After test oracle, the dominant problem is test generation, followed by engineering aspects of assertions. Solutions are typically embedded in tool prototypes and evaluated throughout a limited number of cases, whereas using large-scale industrial settings is still a noticeable method. We conclude that assertions would be worth more attention in future research, particularly regarding the new and emerging demands (e.g., verification of programs with uncertainty), for effective, applicable, and domain-specific solutions.