  Any subanalytic germ $(X,0) \subset (\mathbb R^n,0)$ is equipped with two
natural metrics: its outer metric, induced by the standard Euclidean metric of
the ambient space, and its inner metric, which is defined by measuring the
shortest length of paths on the germ $(X,0)$. The germs for which these two
metrics are equivalent up to a bilipschitz homeomorphism, which are called
Lipschitz Normally Embedded, have attracted a lot of interest in the last
decade. In this survey we discuss many general facts about Lipschitz Normally
Embedded singularities, before moving our focus to some recent developments on
criteria, examples, and properties of Lipschitz Normally Embedded complex
surfaces. We conclude the manuscript with a list of open questions which we
believe to be worth of interest.
