The emergence of infections that are resistant to antibiotics presents a major current global health challenge. One way to address this challenge is to understand better how antibiotics work, in order to devise more effective strategies for using them. I will present a joint experimental – theoretical study in which we investigate how the efficacy of antibiotic treatment depends on whether bacteria are growing fast or slowly, with possible relevance to treating chronic versus virulent infections. I will also describe theoretical work in which we investigate the effect of spatial gradients of antibiotic on the rate of evolution of resistant bacteria.