There has long been a fruitful interaction between model theory and descriptive set theory, particularly regarding the theory of Polish groups. In my talk I will discuss applications of continuous logic (a version of first-order logic adapted to the setting of separable metric spaces rather than countable discrete structures) to the theory of Polish groups. I’ll try to discuss examples where this has been fruitful (some new automatic continuity results, a continuous version of the small index property) as well as an example related to Ramsey theory and extreme amenability where continuous logic might be useful but the discrete setting seems essentially different from the continuous one.