A principal challenge in realizing the potential of quantum computing lies in our ability to perform computations fault-tolerantly, in the presence of the noise inherent to quantum devices. Non-Clifford quantum gates, which are analogous to the classical multiplication (i.e. AND) gate, are particularly difficult to implement fault-tolerantly. We show how to perform such gates with significantly lower asymptotic overhead than was achievable with prior techniques.For this purpose, we present new constructions of quantum error-correcting codes supporting transversal non-Clifford gates, meaning that the desired logical gates can be executed by applying a constant-depth physical circuit to the code state. In particular, we present the first asymptotically good such codes, as well as the first such codes with low-weight parity-checks that have almost linear dimension and polynomial distance. Our constructions are based on algebraic codes and expander graphs, which we combine using topological techniques.No prior knowledge of quantum computing will be assumed.Based on joint works with Venkatesan Guruswami and Ting-Chun Lin.