Many problems in mathematical physics involve disorder across many length scales, which is often addressed heuristically using renormalization group arguments. Particularly challenging are ``infrared'' renormalizations, in which small-scale disorder needs to be ``integrated out'' to reveal the macroscopic behavior of the system. In many interesting models, this procedure must be iterated a large (or even infinite) number many times. Finding a way to do this rigorously is a big challenge, and it has been the focus of my research for the last several years. In this talk I will try to explain some of the interesting open problems and the ``coarse-graining theory'' we have been developing for attacking them. A particular application I will discuss is the analysis of stochastic processes exhibiting anomalous diffusion (in which the variance of the particle at time