It is very hard to compute the electrical resistivity of strongly interacting electrons in a dirty metal from first principles. I will discuss recent progress on a theory of transport bounds, which allow us to estimate the resistivity non-perturbatively. I will show that recent conjectures for bounds on diffusion constants are, in general, false. However, in many models -- including resistor network theory, hydrodynamics, AdS/CMT and kinetic theory -- I will provide rigorous and non-trivial bounds on the resistivity by minimizing entropy production. I will compare the predictions of this formalism to experimental puzzles in a broad class of Fermi and non-Fermi liquids.
Although this talk will focus on electrical transport, our techniques straightforwardly generalize to other linear response coefficients.