The theory of averaging and homogenization for SDEs and PDEs with multiple scales  is by now very well developed, and it has also led to the development of efficient numerical techniques. There are instances, however, where the full dynamics (the slow component of which converges to the the homogenized limit) is not known exactly and the coarse-grained dynamics has to be inferred from data. In this talk I will present techniques for carrying out such a program. I will also show how the coarse-grained dynamics can be used in order to study the control problem for the full dynamics. In particular, I will consider three problems: 1. Inference for multiscale diffusions. 2. Inverse problems for elliptic PDEs with rapidly oscillating coefficients. 3. Stochastic optimal control for SDEs with multiple scales.