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Large language model (LLM) based chatbots, such as ChatGPT, have attracted huge interest in foundation models. It is widely believed that foundation models will serve as the fundamental building blocks for future AI systems. However, the architecture design of foundation model based systems has not yet been systematically explored. There is limited understanding about the impact of introducing foundation models in software architecture. Therefore, in this paper, we propose a taxonomy of foundation model based systems, which classifies and compares the characteristics of foundation models and system design options. Our taxonomy comprises three categories: the pretraining and adaptation of foundation models, the architecture design of foundation model based systems, and responsible-AI-by-design. This taxonomy can serve as concrete guidance for designing foundation model based systems.