Consider a matrix $A$ with complex, bounded, coefficients, which satisfies an ellipticity condition. In 1953 Tosio Kato asked what is the domain of the square root operator of $-div(A\nabla)$. Does it coincide with the one of $\nabla$ and do they have comparable $L^2$ norms? For example, this is the case when $A$ is the identity. The problem made history as the Kato square root problem, and was solved in 2002 by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian. In the first part of this talk, I introduce the Kato square root problem and its applications (to boundary value problems, perturbation estimates, boundedness of Cauchy integral on Lipschitz curves). In the second part, I will survey the recent developments aiming to push the techniques used to solve the Kato problem on on the whole space to more general manifolds. A particularly challenging question is if we can allow degenerate ellipticity, meaning that the matrix$ A$ perturbing the operator can fail to be elliptic at some point. This question is phrased in terms of weighted estimates for the operator $-div(A\nabla)$, for which one wants to prove quadratic estimates.