This is a survey on the Teichmüller metric and on Thurston’s asymmetric metric on Teichmüller space. The stress is on the similarities and the differences between the two metrics. The results on Teichmüller's metric that are described in this chapter are classical, but some of the results on Thurston’s asymmetric metric are new. The authors study in particular the global behaviour of the distance functions and on the limiting behaviour of geodesic rays. Stretch lines and anti-stretch lines are studied in some detail. The chapter ends with a commented list of open questions.