In this talk I will review the notion of unbalanced optimal transport, introduced $\sim10$ years ago to handle mass variarions betwenn nonnegative measures. I will discuss the induced pseudo-Riemannian structure and gradient-flow evolution, corresponding to some class of parabolic reaction-diffusion equations. If time permits I will present two applications: a fitness-driven model from population dynamics, and a Hele-Shaw free boundary problem for tumor growth.