Our aim is to study the Total Variation Flow (TVF) in metric random walk spaces (MRWS) which include as particular cases: the TVF on locally finite weighted connected graphs. We introduce the concepts of perimeter and mean curvature for subsets of a MRWS. After proving the existence and uniqueness of solutions of the TVF, we study the asymptotic behaviour of those solutions, and for such aim we establish some inequalities of Poincar ́e type. Furthermore, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the 1-Laplacian operator. In connection with the Cheeger cut problem we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem.