We derive an asymptotic expansion of Laplace-type integrals in which dimension grows together with the large parameter in the exponent. This fills a gap in the theory between the classical fixed dimensional regime dating back to Laplace, and more recent work on the asymptotic expansion of infinite dimensional Laplace-type integrals due to Ben Arous. We also present related work on approximations of high-dimensional Laplace-type probability densities. These results resolve several long-standing open questions in the theory of both general asymptotic analysis and high-dimensional Bayesian statistics. Beyond their theoretical significance, our results are useful for statistical computation such as model selection and uncertainty quantification in Bayesian inverse problems and other data science settings.