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Procedural material graphs are a compact, parameteric, and resolution-independent representation that are a popular choice for material authoring. However, designing procedural materials requires significant expertise and publicly accessible libraries contain only a few thousand such graphs. We present MatFormer, a generative model that can produce a diverse set of high-quality procedural materials with complex spatial patterns and appearance. While procedural materials can be modeled as directed (operation) graphs, they contain arbitrary numbers of heterogeneous nodes with unstructured, often long-range node connections, and functional constraints on node parameters and connections. MatFormer addresses these challenges with a multi-stage transformer-based model that sequentially generates nodes, node parameters, and edges, while ensuring the semantic validity of the graph. In addition to generation, MatFormer can be used for the auto-completion and exploration of partial material graphs. We qualitatively and quantitatively demonstrate that our method outperforms alternative approaches, in both generated graph and material quality.