The interleaving distance for persistence modules, originally defined by Chazal et al., is arguably the most powerful, generalizable mathematical idea to come out of TDA in the last decade. The categorification of persistence modules and the related interleaving distance provided a plug-and-play system to create new metrics for functors with a poset category domain. In this talk, we will generalize this work even further to give a definition of the interleaving distance for a category with a flow; that is, a category C with a functor S:R≥0→End(C) which satisfies certain compatibility conditions. From this framework, we can see that many commonly used metrics, such as the Hausdorff distance on sets and the ℓ∞ distance on \Rn are all examples of interleaving distances. The categorical viewpoint gives an immediate construction for a host of stability theorems. Further, a new construction on elements of a category with a flow called a hom-tree provides a lower bound for the interleaving distance.