I will introduce definable dimension functions (in the axiomatic sense of L. van den Dries), and then, after briefly reviewing some familiar results for o-minimal structures, survey some newer results on such dimensions in ordered structures with derivations. I will focus on how the relevant dimensions are connected to the order topology, mentioning three sets of results. First, results of M. Aschenbrenner, L. van den Dries, and J. van der Hoeven for the dimension coming from differential-algebraic closure in closed \(H\)-fields; examples of closed \(H\)-fields include logarithmic-exponential transseries, surreal numbers, and maximal Hardy fields. Second, my analogous results for tame (i.e., Dedekind complete) pairs of closed \(H\)-fields, but for which the dimension must be modified. Third, results of joint work with A. Gehret and E. Kaplan for the asymptotic couple of the differential field of logarithmic transseries, which is the value group of that differential field equipped with the map induced by its logarithmic derivative. The latter results are an update to part of Gehret's Model Theory Seminar of 2024.12.11. Nevertheless, the talk will be mostly self-contained.