This paper provides the first transcription and English translation of Galileo's "Theorica speculi concavi spha[e]rici", because of its importance as being almost the only source of Galileo's early optics. The introduction to the transcription and translation has limited aims. It provides an overview of the "Theorica" showing its dependence on, and surpassing of medieval catoptrics and suggests possible links with experiments with projecting devices that might have been useful to Galileo.