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A piecewise flat Finsler metric on a triangulated surface $M$ is a metricwhose restriction to any triangle is a flat triangle in some Minkowski spacewith straight edges. One of the main purposes of this work is to study theproperties of geodesics on a piecewise flat Finsler surface, especially when itmeets a vertex. Using the edge-crossing equation, we define two classes ofpiecewise flat Finsler surfaces, namely, Landsberg type and Berwald type. Wededuce an explicit condition for a geodesic to be extendable at a vertex, anddefine the curvature which measures the \textit{amount} of such extensions. Thedependence of the curvature on an incoming or outgoing tangent directioncorresponds to the feature of flag curvature in Finsler geometry. When thepiecewise flat Finsler surface is of Landsberg type, the curvature is onlyrelevant to the vertex, and we prove a combinatoric Gauss-Bonnet formula whichgeneralizes both the Gauss-Bonnet formulas for piecewise flat Riemannianmanifolds and for smooth Landsberg surfaces.

Geodesic and curvature of piecewise flat Finsler surfaces