State-of-the-art generalist manipulation policies have enabled the deployment of robotic manipulators in unstructured human environments. However, these frameworks struggle in cluttered environments primarily because they utilize auxiliary modules for low-level motion planning and control. Motion planning remains challenging due to the high dimensionality of the robot's configuration space and the presence of workspace obstacles. Neural motion planners have enhanced motion planning efficiency by offering fast inference and effectively handling the inherent multi-modality of the motion planning problem. Despite such benefits, current neural motion planners often struggle to generalize to unseen, out-of-distribution planning settings. This paper reviews and analyzes the state-of-the-art neural motion planners, highlighting both their benefits and limitations. It also outlines a path toward establishing generalist neural motion planners capable of handling domain-specific challenges. For a list of the reviewed papers, please refer to https://davoodsz.github.io/planning-manip-survey.github.io/.