This research work address the moving object detection and tracking domain, describing some mathematical models that have been appplied successfully in this computer vision field. Thus, several variational and non-variational partial differential equation (PDE)-based models for image and video object detection and tracking are surveyed here. The detection and tracking techniques based on Geometric Active Contour models, also called snakes, which represent energy-based (variational) image segmentation schemes, are presented first. Then, another category of PDE-based geometric models for object detection and tracking, containing mathematical models based on level-set functions, is disscused here. Moving object tracking approaches based on the video optical flow that is estimated using PDE-based models are described next. The histogram-based PDE models for video object tracking are then presented. Image object detection techniques using PDE-based boundary and contour extraction models are also discussed. Finally, our own contributions in this research domain are described here. Thus, some nonlinear PDE-based automatic detection and tracking frameworks for certain classes of video objects, such as pedestrians and vehicles, are briefly presented.