Discovering new materials can have significant scientific and technological implications but remains a challenging problem today due to the enormity of the chemical space. Recent advances in machine learning have enabled data-driven methods to rapidly screen or generate promising materials, but these methods still depend heavily on very large quantities of training data and often lack the flexibility and chemical understanding often desired in materials discovery. We introduce LLMatDesign, a novel language-based framework for interpretable materials design powered by large language models (LLMs). LLMatDesign utilizes LLM agents to translate human instructions, apply modifications to materials, and evaluate outcomes using provided tools. By incorporating self-reflection on its previous decisions, LLMatDesign adapts rapidly to new tasks and conditions in a zero-shot manner. A systematic evaluation of LLMatDesign on several materials design tasks, in silico, validates LLMatDesign's effectiveness in developing new materials with user-defined target properties in the small data regime. Our framework demonstrates the remarkable potential of autonomous LLM-guided materials discovery in the computational setting and towards self-driving laboratories in the future.