  We study the Neumann and Dirichlet problems for the total variation flow in
metric measure spaces. We prove existence and uniqueness of weak solutions and
study their asymptotic behaviour. Furthermore, in the Neumann problem we
provide a notion of solutions which is valid for $L^1$ initial data, as well as
prove their existence and uniqueness. Our main tools are the first-order linear
differential structure due to Gigli and a version of the Gauss-Green formula.
