Comprehensive understanding of dynamic scenes is a critical prerequisite for intelligent robots to autonomously operate in their environment. Research in this domain, which encompasses diverse perception problems, has primarily been focused on addressing specific tasks individually rather than modeling the ability to understand dynamic scenes holistically. In this paper, we introduce a novel perception task denoted as multi-object panoptic tracking (MOPT), which unifies the conventionally disjoint tasks of semantic segmentation, instance segmentation, and multi-object tracking. MOPT allows for exploiting pixel-level semantic information of thing and stuff classes, temporal coherence, and pixel-level associations over time, for the mutual benefit of each of the individual sub-problems. To facilitate quantitative evaluations of MOPT in a unified manner, we propose the soft panoptic tracking quality (sPTQ) metric. As a first step towards addressing this task, we propose the novel PanopticTrackNet architecture that builds upon the state-of-the-art top-down panoptic segmentation network EfficientPS by adding a new tracking head to simultaneously learn all sub-tasks in an end-to-end manner. Additionally, we present several strong baselines that combine predictions from state-of-the-art panoptic segmentation and multi-object tracking models for comparison. We present extensive quantitative and qualitative evaluations of both vision-based and LiDAR-based MOPT that demonstrate encouraging results.