Wheeled mobile robots (WMRs) have been a focus of research for several decades, particularly concerning navigation strategies in static and dynamic environments. This review article carefully examines the extensive academic efforts spanning several decades addressing navigational complexities in the context of WMR route analysis. Several approaches have been explored by various researchers, with a notable emphasis on the inclusion of stability and intelligent capabilities in WMR controllers attracting the attention of the academic community. This study traces historical and contemporary WMR research, including the establishment of kinetic stability and the construction of intelligent WMR controllers. WMRs have gained prominence in various applications, with precise navigation and efficient control forming the basic prerequisites for their effective performance. The review presents a comprehensive overview of stability analysis and navigation techniques tailored for WMRs. Initially, the exposition covers the basic principles of WMR dynamics and kinematics, explaining the different wheel types and their associated constraints. Subsequently, various stability analysis approaches, such as Lyapunov stability analysis and passivation-based control, are discussed in depth in the context of WMRs. Starting an exploration of navigation techniques, the review highlights important aspects including path planning and obstacle avoidance, localization and mapping, and trajectory tracking. These techniques are carefully examined in both indoor and outdoor settings, revealing their benefits and limitations. Finally, the review ends with a comprehensive discussion of the current challenges and possible routes in the field of WMR. The discourse includes the fusion of advanced sensors and state-of-the-art control algorithms, the cultivation of more robust and reliable navigation strategies, and the continued exploration of novel WMR applications. This article also looks at the progress of mobile robotics during the previous three decades. Motion planning and path analysis techniques that work with single and multiple mobile robots have been discussed extensively. One common theme in this research is the use of soft computing methods to give mobile robot controllers cognitive behaviors, such as artificial neural networks (ANNs), fuzzy logic control (FLC), and genetic algorithms (GAs). Nevertheless, there is still a dearth of applications for mobile robot navigation that leverage nature-inspired algorithms, such as firefly and ant colony algorithms. Remarkably, most studies have focused on kinematics analysis, with a small number also addressing dynamics analysis.