Motivated by a Gan-Loh-Sudakov-type problem, we introduce the regular Turán numbers, a natural variation on the classical Turán numbers for which the host graph is required to be regular. Among other results, we prove a striking supersaturation version of Mantel's theorem in the case of a regular host graph of odd order. We also characterise the graphs for which the regular Turán numbers behave classically or otherwise.