We introduce Hi-COLA, a code designed to run fast, approximate N-body simulations of non-linear structure formation in reduced Horndeski gravity. Given an input Lagrangian, Hi-COLA dynamically constructs the appropriate field equations and consistently solves for the cosmological background, linear growth, and screened fifth force of that theory. Hence Hi-COLA is a general, adaptable, and useful tool that allows the mildly non-linear regime of many Horndeski theories to be investigated for the first time, at low computational cost. In this work, we first describe the screening approximations and simulation setup of Hi-COLA for theories with Vainshtein screening. We validate the code against traditional N-body simulations for cubic Galileon gravity, finding 2.5% agreement up to k max = 1.2 h/Mpc. To demonstrate the flexibility of Hi-COLA, we additionally run the first simulations of an extended shift-symmetric gravity theory. We use the consistency and modularity of Hi-COLA to dissect how the modified background, linear growth, and screened fifth force all contribute to departures from ΛCDM in the non-linear matter power spectrum. Hi-COLA can be found at https://github.com/Hi-COLACode/Hi-COLA.

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