ABSTRACT In this paper, we initiate the study of discrepancy questions for combinatorial designs. Specifically, we show that, for every fixed and , any ‐colouring of the triples on admits a Steiner triple system of order with discrepancy . This is not true for , but we are able to asymptotically characterise all 2‐colourings which do not contain a Steiner triple system with high discrepancy. The key step in our proofs is a characterisation of 3‐uniform hypergraphs avoiding a certain natural type of induced subgraphs, contributing to the structural theory of hypergraphs.