This chapter is dedicated to the study of the total variation flow on the whole space. In order to use the Anzellotti pairings and the Gauss-Green formula introduced in Chapter 4, we suppose that the metric space is complete, separable, equipped with a doubling measure, and that it supports a weak Poincaré-type inequality. The total variation flow in this setting is understood as the gradient flow of the 1-Cheeger energy, which (as shown by Ambrosio and Di Marino) is convex and lower semicontinuous with respect to L2 convergence; then, the Brezis–Komura theorem guarantees the existence of a unique strong solution. We now provide a characterisation of the subdifferential of the 1-Cheeger energy, introduce the notion of weak solution to the total variation flow based on this characterisation, and prove their existence and uniqueness. We also comment on the asymptotic behaviour of weak solutions and introduce a notion of entropy solutions for initial data in L1 under the assumption that the measure of the space is finite.