Quasi-Borel spaces are a new mathematical structure that supports higher-order probability theory, first-order iteration, and modular semantic validation of Bayesian inference algorithms with continuous distributions. Like a measurable space, a quasi-Borel space is a set with extra structure suitable for defining probability and measure distributions. But unlike measurable spaces, quasi-Borel spaces and their structure- preserving maps form a well-behaved category: they are cartesian- closed, and so suitable for higher-order semantics, and they also form a model of Kock’s synthetic measure theory, and so suitable for probabilistic, and measure-theoretic, developments, such as the Metropolis- Hastings-Green theorem underlying Markov-Chain Monte-Carlo algorithms.