Mobile robots have been making a significant contribution to the advancement of many sectors including automation of mining, space, surveillance, military, health, agriculture and many more. Safe and efficient navigation is a fundamental requirement of mobile robots, thus, the demand for advanced algorithms rapidly increased. Mobile robot navigation encompasses the following four requirements: perception, localization, pathplanning and motion control. Among those, path-planning is a vital part of a fast, secure operation. During the last couple of decades, many path-planning algorithms were developed. Despite most of the mobile robot applications being in dynamic environments, the number of algorithms capable of navigating robots in dynamic environments is limited. This paper presents a qualitative comparative study of the up-to-date mobile robot pathplanning methods capable of navigating robots in dynamic environments. The paper discusses both classical and heuristic methods including artificial potential field, genetic algorithm, fuzzy logic, neural networks, artificial bee colony, particle swarm optimization, bacterial foraging optimization, ant-colony and Agoraphilic algorithm. The general advantages and disadvantages of each method are discussed. Furthermore, the commonly used state-of-the-art methods are critically analyzed based on six performance criteria: algorithm's ability to navigate in dynamically cluttered areas, moving goal hunting ability, object tracking ability, object path prediction ability, incorporating the obstacle velocity in the decision, validation by simulation and experimentation. This investigation benefits researchers in choosing suitable path-planning methods for different applications as well as identifying gaps in this field.