This paper addresses non-Gaussian regression with neural networks via the use of the Tukey g-and-h distribution.The Tukey g-and-h transform is a flexible parametric transform with two parameters $g$ and $h$ which, when applied to a standard normal random variable, introduces both skewness and kurtosis, resulting in a distribution commonly called the Tukey g-and-h distribution. Specific values of $g$ and $h$ produce good approximations to other families of distributions, such as the Cauchy and student-t distributions. The flexibility of the Tukey g-and-h distribution has driven its popularity in the statistical community, in applied sciences and finance. In this work we consider the training of a neural network to predict the parameters of a Tukey g-and-h distribution in a regression framework via the minimization of the corresponding negative log-likelihood, despite the latter having no closed-form expression. We demonstrate the efficiency of our procedure in simulated examples and apply our method to a real-world dataset of global crop yield for several types of crops. Finally, we show how we can carry out a goodness-of-fit analysis between the predicted distributions and the test data. A Pytorch implementation is made available on Github and as a Pypi package.