With the increasing efforts to utilize information and communication technologies (ICT) in disaster management, the massive amount of heterogeneous data that is generated through ubiquitous sensors paves the way for fast and informed decisions in the case of disasters. Utilization of the big “sensed” data leads to an effective and efficient management of disaster situations so as to prevent human and economic losses. The advancement of built-in sensing technologies in smart mobile devices enables crowdsourcing of sensed data, which is known as mobile crowdsensing (MCS). This systematic literature review investigates the use of mobile crowdsensing in disaster management on the basis of the built-in sensor types in smart mobile devices, disaster management categories, and the disaster management cycle phases (i.e., mitigation, preparedness, response, and recovery activities). Additionally, this work seeks to unveil the frameworks or models that can potentially guide disaster management authorities towards integrating crowd-sensed data with their existing decision-support systems. The vast majority of the existing studies are conceptual as they highlight a challenge in experimental testing of the disaster management solutions in real-life settings, and there is little emphasis on the use cases of crowdsensing through smartphone sensors in disaster incidents. In light of a thorough review, we provide and discuss future directions and open issues for mobile crowdsensing-aided disaster management.