  In this paper we study the Total Variation Flow (TVF) in metric random walk
spaces, which unifies into a broad framework the TVF on locally finite weighted
connected graphs, the TVF determined by finite Markov chains and some nonlocal
evolution problems. Once the existence and uniqueness of solutions of the TVF
has been proved, we study the asymptotic behaviour of those solutions and, with
that aim in view, we establish some inequalities of Poincar\'{e} type. In
particular, for finite weighted connected graphs, we show that the solutions
reach the average of the initial data in finite time. Furthermore, we introduce
the concepts of perimeter and mean curvature for subsets of a metric random
walk space and we study the relation between isoperimetric inequalities and
Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and
calibrable sets in metric random walk spaces and characterize calibrability by
using the $1$-Laplacian operator. Finally, we study the eigenvalue problem
whereby we give a method to solve the optimal Cheeger cut problem.
