The theory of Hamilton-Jacobi equations in Hilbert and some
Banach spaces is relatively well developed. Much less is known about equations in spaces of measures, and more general metric spaces. We will present a notion of metric viscosity solution which applies to a class of Hamilton-Jacobi equations in geodesic metric spaces and gives well posedness for such equations. We will also discuss other approaches to Hamilton-Jacobi equations in metric spaces, in particular in the Wasserstein space, and discuss some applications of such equations.