The authors study two collections of topologies on the set \(C(X)\) of continuous real-valued functions on \(X\): the set-open topologies and the uniform topologies, both over certain families of bounded sets in \(X\) (including the families of all finite sets, compact sets, countably compact sets, pseudocompact sets, and bounded sets). A detailed comparison of these topologies is made, and a number of examples are given. Also the properties of separable, metrizable, and complete are characterized for the uniform topologies in general.