  These are notes on the hyperbolic geometry of surfaces, Teichm{\"u}ller
spaces and Thurston's metric on these spaces. They are associated with lectures
I gave at the Morningside Center of Mathematics of the Chinese Academy of
Sciences in March 2019 and at the Chebyshev Laboratory of the Saint Petersburg
State University in May 2019. In particular, I survey several results on the
behavior of stretch lines, a distinguished class of geodesics for Thurston's
metric and I point out several analogies between this metric and
Teichm{\"u}ller's metric. Several open questions are addressed. The final
version of these notes will appear in the book "Moduli Spaces and Locally
Symmetric Spaces", edited by L. Ji and S.-T. Yau, International Press, 2021.
