Recommender systems (RSs) have been playing an increasingly important role for informed consumption, services, and decision-making in the overloaded information era and digitized economy. In recent years, session-based recommender systems (SBRSs) have emerged as a new paradigm of RSs. Different from other RSs such as content-based RSs and collaborative filtering-based RSs that usually model long-term yet static user preferences, SBRSs aim to capture short-term but dynamic user preferences to provide more timely and accurate recommendations sensitive to the evolution of their session contexts. Although SBRSs have been intensively studied, neither unified problem statements for SBRSs nor in-depth elaboration of SBRS characteristics and challenges are available. It is also unclear to what extent SBRS challenges have been addressed and what the overall research landscape of SBRSs is. This comprehensive review of SBRSs addresses the above aspects by exploring in depth the SBRS entities (e.g., sessions), behaviours (e.g., users’ clicks on items), and their properties (e.g., session length). We propose a general problem statement of SBRSs, summarize the diversified data characteristics and challenges of SBRSs, and define a taxonomy to categorize the representative SBRS research. Finally, we discuss new research opportunities in this exciting and vibrant area.