Accurate cell-type annotation from spatially resolved single cells is crucial to understand functional spatial biology that is the basis of tissue organization. However, current computational methods for annotating spatially resolved single-cell data are typically based on techniques established for dissociated single-cell technologies and thus do not take spatial organization into account. Here we present STELLAR, a geometric deep learning method for cell-type discovery and identification in spatially resolved single-cell datasets. STELLAR automatically assigns cells to cell types present in the annotated reference dataset and discovers novel cell types and cell states. STELLAR transfers annotations across different dissection regions, different tissues and different donors, and learns cell representations that capture higher-order tissue structures. We successfully applied STELLAR to CODEX multiplexed fluorescent microscopy data and multiplexed RNA imaging datasets. Within the Human BioMolecular Atlas Program, STELLAR has annotated 2.6 million spatially resolved single cells with dramatic time savings.