Neuro-Symbolic Diffusion (NSD) introduced as a framework to generate samples under physical, structural, and operational constraints.Combines diffusion steps with symbolic optimization for generating consistent outputs adhering to user-defined constraints.NSD supports continuous (e.g., images, trajectories) and discrete (e.g., molecular structures, natural language) output generation.Demonstrated applications include safety, data scarcity, and out-of-domain generalization challenges.