In recent years there has been considerable activity in generalizing to continuous settings techniques from logic (set theory and model theory) that initially were devised for discrete settings. Several model-theoretic frameworks have been proposed as formalisms for the continuous setting (Chang-Keisler, Henson, Ben-Yaacov); however, they have turned out to be equivalent. I will state a maximality theorem that, among other things, characterizes such frameworks. The characterization is in terms of omitting types, and applies not only to the already proposed formalisms, but to a wide range of logics, namely, logics for which certain topologies are uniformizable.