Apocalyptic projections of the impact of antimicrobial resistance (AMR) on human well-being project 100 trillion dollars in lost GDP and 10 million annual deaths from a "do nothing scenario," one which makes the assumption that resistance to all drugs will reach 100% over the next 15 years. Fortunately, such a scenario is pure science fiction. But why? Empirical data suggest that the fraction of bacteria resistant to a given antibiotic class is roughly proportional to national or regional antimicrobial use, but that it doesn't change dramatically over time. Evolutionary theory, on the other hand, suggests that selection (here, antibiotic use) should determine the rate of change, rather than the steady-state level, of the selected trait (here, resistance). This is the "puzzle of coexistence" of drug-resistant and susceptible bacteria, one that has been bothering a small group of collaborators for about a decade now. This talk will describe the progress made in resolving the puzzle, focusing on Streptococcus pneumoniae as a model organism, with the following themes: 1) Demonstrating the phenomenon exists. 2) Rejecting as mathematical tricks the simple models that predict "coexistence for free." 3) Describing the failure of non-tricky models to reproduce the phenomenon. 4) Describing the contribution of various forms of heterogeneity in the host population, captured in an agent-based model, to partially resolving the puzzle. 5) Describing the contribution of negative frequency-dependent selection (eg on antigens) combined with epistasis affecting the fitness of resistance genes to partially resolving the puzzle. 6) Describing the evidence that negative frequency-dependent selection is pervasive in S. pneumoniae. 7) Identifying remaining open questions. 8) Describing some preliminary work to test one of the key assumptions of AMR models.