A survey on Lipschitz-free Banach Spaces

This article is a survey of Lipschitz\dywiz free Banach spaces and recent progress in the understanding of their structure. The results we present have been obtained in the last fifteen years (and quite often in the last five years). We give a self\dywiz contained presentation of the basic properties of Lipschitz\dywiz free Banach spaces and investigate some specific topics: non-linear transfer of asymptotic smoothness, approximation properties, norm\dywiz attainment. Section 5 consists mainly of unpublished results. A list of open problems with comentary is provided.