The intestine performs functions central to human health by breaking down food and absorbing nutrients while maintaining a selective barrier against the intestinal microbiome. Key to this barrier function are the combined efforts of lumen-lining specialized intestinal epithelial cells, and the supportive underlying immune cell-rich stromal tissue. The discovery that the intestinal epithelium can be reproduced in vitro as intestinal organoids introduced a new way to understand intestinal development, homeostasis, and disease. However, organoids reflect the intestinal epithelium in isolation whereas the underlying tissue also contains myriad cell types and impressive chemical and structural complexity. This review dissects the cellular and matrix components of the intestine and discusses strategies to replicate them in vitro using principles drawing from bottom-up biological self-organization and top-down bioengineering. It also covers the cellular, biochemical and biophysical features of the intestinal microenvironment and how these can be replicated in vitro by combining strategies from organoid biology with materials science. Particularly accessible chemistries that mimic the native extracellular matrix are discussed, and bioengineering approaches that aim to overcome limitations in modelling the intestine are critically evaluated. Finally, the review considers how further advances may extend the applications of intestinal models and their suitability for clinical therapies.