We propose and analyze a general model of priority races. Researchers privately have breakthroughs and decide how long to let their ideas mature before disclosing them, thereby establishing priority. Two-researcher, symmetric priority races have a unique equilibrium that can be characterized by a differential equation. We study how the shape of the breakthrough distribution and of the returns to maturation affect maturation delays and research quality, both in dynamic and comparative-statics analyses. Making researchers better at discovering new ideas or at developing them has contrasted effects on research quality. Being closer to the technological frontier enhances the value of maturation for researchers, which mitigates the negative impact on research quality of the race for priority. Finally, when researchers differ in their abilities to do creative work or in the technologies they use to develop their ideas, more efficient researchers always let their ideas mature more than their less efficient opponents. Our theoretical results shed light on academic competition, patent races, and innovation quality.