Much recent research on information retrieval has focused on how to transfer from one task (typically with abundant supervised data) to various other retrieval tasks where supervision is limited, with the implicit assumption that it is possible to generalize from one task to all the rest. However, this overlooks the fact that there are many diverse and unique retrieval problems, each targeting different search intents, queries, and search domains. In this paper, we suggest to work on Few-shot Dense Retrieval, a setting where each task comes with a short description and a few examples. To address this, we introduce Prompt-based Query Generation forRetrieval (Promptagator): for each task, we feed the few-shot examples to a large language model (LLM) and prompt it to behave as a task-specific query generator. Using this, we can synthetically generate a large number of relevant queries for any document, yielding abundant data for training task-specific retrievers --- with no reliance on traditional resources such as Natural Questions (Kwiatkowskiet al., 2019) or MS MARCO (Nguyen et al., 2016). Surprisingly, Promptagator with only 8 annotated examples enables efficient dual encoder retrievers to outperform computationally more expensive models trained on MS MARCO such as ColBERT v2 (Santhanam et al., 2022) by more than 1.2 points nDCG@10 on average on 11 retrieval sets. Further training standard-size re-rankers using the same generated data yields another 5.0 points nDCG@10 improvement. Our studies show that synthetic query generation can be far more effective than previously observed, especially when a small amount of task-specific knowledge is given.