  In this paper we develop an approach to conformal geometry of piecewise flat
metrics on manifolds. In particular, we formulate the combinatorial Yamabe
problem for piecewise flat metrics. In the case of surfaces, we define the
combinatorial Yamabe flow on the space of all piecewise flat metrics associated
to a triangulated surface. We show that the flow either develops removable
singularities or converges exponentially fast to a constant combinatorial
curvature metric. If the singularity develops, we show that the singularity is
always removable by a surgery procedure on the triangulation.
  We conjecture that after finitely many such surgery changes on the
triangulation, the flow converges to the constant combinatorial curvature
metric as time approaches infinity.
