The recent discovery of the Higgs boson at the Large Hadron Collider represents a triumph for the Standard Model of particle physics. However, from a theorist's perspective several puzzles remain, among them the stability of the Higgs mass and the identity of dark matter. Models in which the Higgs itself is a composite bound state can provide elegant solutions to these puzzles, but these composite systems are inherently strongly coupled, and as such they can be challenging to investigate. Numerical field theory, which has enjoyed enormous success as a tool for study of the strong nuclear force, can provide new insights and deeper understanding. In this talk, I will briefly discuss the most promising models of Higgs compositeness, and present recent large-scale simulation results targeting such models, with an emphasis on connections to experiment.