Nearby cells within tissues communicate through ligand-receptor signaling interactions. Emerging spatial transcriptomic technologies provide a tremendous opportunity to systematically detect ligand-receptor signaling, but no method operates at cellular resolution in the spatial context. We developed CytoSignal to infer the locations and dynamics of cell-cell communication at cellular resolution from spatial transcriptomic data. CytoSignal is based on the simple insight that signaling is a protein-protein interaction that occurs at a specific tissue location when ligand and receptor are expressed in close spatial proximity. Our cellular-resolution, spatially-resolved signaling scores allow several novel types of analyses: we identify spatial gradients in signaling strength; separately quantify the locations of contact-dependent and diffusible interactions; and detect signaling-associated differentially expressed genes. Additionally, we can predict the temporal dynamics of a signaling interaction at each spatial location. CytoSignal is compatible with nearly every kind of spatial transcriptomic technology including FISH-based protocols and spot-based protocols without deconvolution. We experimentally validate our results in situ by proximity ligation assay, confirming that CytoSignal scores closely match the tissue locations of ligand-receptor protein-protein interactions. Our work addresses the field's current need for a robust and scalable tool to detect cell-cell signaling interactions and their dynamics at cellular resolution from spatial transcriptomic data.