I will introduce two new geometric evolution equations and discuss their applications. First, I will show a "sphere theorem" in four dimensions using the gradient flow of the L^2 norm of the curvature tensor. Then I will discuss a new geometric evolution equation generalizing the Kahler Ricci flow onto certain non-Kahler manifolds. I will exhibit a remarkable relationship between this flow and the B-field renormalization group flow of string theory, and describe how some reasonable conjectures for this flow can be used to understand the long unsolved problem of the classification of the mysterious Class VII surfaces.