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Open-vocabulary object detection, which is concerned with the problem of detecting novel objects guided by natural language, has gained increasing attention from the community. Ideally, we would like to extend an open-vocabulary detector such that it can produce bounding box predictions based on user inputs in form of either natural language or exemplar image. This offers great flexibility and user experience for human-computer interaction. To this end, we propose a novel open-vocabulary detector based on DETR -- hence the name OV-DETR -- which, once trained, can detect any object given its class name or an exemplar image. The biggest challenge of turning DETR into an open-vocabulary detector is that it is impossible to calculate the classification cost matrix of novel classes without access to their labeled images. To overcome this challenge, we formulate the learning objective as a binary matching one between input queries (class name or exemplar image) and the corresponding objects, which learns useful correspondence to generalize to unseen queries during testing. For training, we choose to condition the Transformer decoder on the input embeddings obtained from a pre-trained vision-language model like CLIP, in order to enable matching for both text and image queries. With extensive experiments on LVIS and COCO datasets, we demonstrate that our OV-DETR -- the first end-to-end Transformer-based open-vocabulary detector -- achieves non-trivial improvements over current state of the arts.