Humans are able to learn to recognize new objects even from a few examples. In contrast, training deep-learning-based object detectors requires huge amounts of annotated data. To avoid the need to acquire and annotate these huge amounts of data, few-shot object detection (FSOD) aims to learn from few object instances of new categories in the target domain. In this survey, we provide an overview of the state of the art in FSOD. We categorize approaches according to their training scheme and architectural layout. For each type of approach, we describe the general realization as well as concepts to improve the performance on novel categories. Whenever appropriate, we give short takeaways regarding these concepts in order to highlight the best ideas. Eventually, we introduce commonly used datasets and their evaluation protocols and analyze the reported benchmark results. As a result, we emphasize common challenges in evaluation and identify the most promising current trends in this emerging field of FSOD.