In this talk I will discuss a new model for random simplicial complexes. Unlike other models, which are generically simply-connected when sufficiently dense, the complexes in this model generically have fundamental group $\mathbb{Z}/2\mathbb{Z}$. I’ll describe results on the asymptotic behavior of the homology and homotopy groups in these complexes as well as how these results imply a “random Borsuk–Ulam theorem”. This talk is based on joint work with Florian Frick.