Abstract. In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to get a boundary for the space of flat metrics. The geometric interpretation is that flat metrics degenerate to mixed structures on the surface: part flat metric and part measured foliation.

We provide a combinatorial condition characterizing curves that are short along a Teichmüller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S ×R are also short in the corresponding Teichmüller geodesic, and we provide examples demonstrating that the converse is not true.

We study how the length and the twisting parameter of a curve change along a Teichmüller geodesic. We then use our results to provide a formula for the Teichmüller distance between two hyperbolic metrics on a surface, in terms of the combinatorial complexity of curves of bounded lengths in these two metrics.

We review and organize some results describing the behavior of a Teichmüller geodesic and draw several applications: 1) We show that Teichmüller geodesics do not back track. 2) We show that a Teichmüller geodesic segment whose endpoints are in the thick part has the fellow travelling property. This fails when the endpoints are not necessarily in the thick part. 3) We show that if an edge of a Teichmüller geodesic triangle passes through the thick part, then it is close to one of the other edges.

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