Estimating hidden states in dynamical systems, also known as optimal filtering, is a long-standing problem in various fields of science and engineering. In this paper, we introduce a general filtering framework, \textbf{LLM-Filter}, which leverages large language models (LLMs) for state estimation by embedding noisy observations with text prototypes. In various experiments for classical dynamical systems, we find that first, state estimation can significantly benefit from the reasoning knowledge embedded in pre-trained LLMs. By achieving proper modality alignment with the frozen LLM, LLM-Filter outperforms the state-of-the-art learning-based approaches. Second, we carefully design the prompt structure, System-as-Prompt (SaP), incorporating task instructions that enable the LLM to understand the estimation tasks. Guided by these prompts, LLM-Filter exhibits exceptional generalization, capable of performing filtering tasks accurately in changed or even unseen environments. We further observe a scaling-law behavior in LLM-Filter, where accuracy improves with larger model sizes and longer training times. These findings make LLM-Filter a promising foundation model of filtering.