The number of research publications dealing with the simultaneous localization and mapping problem has grown significantly over the past 15 years. Many fundamental and practical aspects of simultaneous localization and mapping have been addressed, and some efficient algorithms and practical solutions have been demonstrated. The aim of this paper is to provide a critical review of current theoretical understanding of the fundamental properties of the SLAM problem, such as observability, convergence, achievable accuracy and consistency. Recent research outcomes associated with these topics are briefly discussed together with potential future research directions.