  We investigate the properties of the combinatorial Ricci flow for surfaces,
both forward and backward -- existence, uniqueness and singularities formation.
We show that the positive results that exist for the smooth Ricci flow also
hold for the combinatorial one and that, moreover, the same results hold for a
more general, metric notion of curvature. Furthermore, using the metric
curvature approach, we show the existence of the Ricci flow for polyhedral
manifolds of piecewise constant curvature. We also study the problem of the
realizability of the said flow in $\mathbb{R}^3$.
