The aghq package for implementing approximate Bayesian inference using adaptive quadrature is introduced. The method and software are described, and use of the package in making approximate Bayesian inferences in several challenging low- and high-dimensional models is illustrated. Examples include an infectious disease model; an astrostatistical model for estimating the mass of the Milky Way; two examples in non-Gaussian model-based geostatistics including one incorporating zero-inflation which is not easily fit using other methods; and a model for zero-inflated, overdispersed count data. The aghq package is especially compatible with the popular TMB interface for automatic differentiation and Laplace approximation, and existing users of that software can make approximate Bayesian inferences with aghq using very little additional code. The aghq package is available from CRAN and complete code for all examples in this paper can be found at https://github.com/awstringer1/aghq-software-paper-code.