As the most fundamental tasks of computer vision, object detection and segmentation have made tremendous progress in the deep learning era. Due to the expensive manual labeling, the annotated categories in existing datasets are often small-scale and pre-defined, i.e., state-of-the-art detectors and segmentors fail to generalize beyond the closed-vocabulary. To resolve this limitation, the last few years have witnessed increasing attention toward Open-Vocabulary Detection (OVD) and Segmentation (OVS). In this survey, we provide a comprehensive review on the past and recent development of OVD and OVS. To this end, we develop a taxonomy according to the type of task and methodology. We find that the permission and usage of weak supervision signals can well discriminate different methodologies, including: visual-semantic space mapping, novel visual feature synthesis, region-aware training, pseudo-labeling, knowledge distillation-based, and transfer learning-based. The proposed taxonomy is universal across different tasks, covering object detection, semantic/instance/panoptic segmentation, 3D scene and video understanding. In each category, its main principles, key challenges, development routes, strengths, and weaknesses are thoroughly discussed. In addition, we benchmark each task along with the vital components of each method. Finally, several promising directions are provided to stimulate future research.