Time-change equations are a generalization of ordinary differential equations which are driven by the random, irregular, and possibly densely discontinuous sample paths of the typical stochastic process.  They can be thought of as a multiparametric version of the method of time-changes. Time-change equations can lead to deep results on weak existence and uniqueness of stochastic differential equations and posses a robust strong approximation theory. However, time-change equations are not restricted to Markovian or semimartingale settings. In this talk, we will go through some examples of time-change equations which can be succesfully analyzed (such as (multidimensional) affine processes or sticky Lévy processes) as well as some open problems they suggest.