The EFT bootstrap puts rigorous bounds on Wilson coefficients using dispersion relations derived from the 2-to-2 scattering amplitude. Specifically, it generalises classical positivity bounds to two-sided bounds on ratios of Wilson coefficients. Involving several ratios, we can carve out an allowed region in parameter space and try to explain its shape by looking at the extremal amplitudes saturating the bound. In this talk, I will give a review of the framework and discuss some recent work, including bounds on the EFT of a single massive vector field.