  We present here basic results in Lipschitz Geometry of semialgebraic surface
germs. Although bi-Lipschitz classification problem of surface germs with
respect to the inner metric was solved long ago, classification with respect to
the outer metric remains an open problem. We review recent results related to
the outer and ambient bi-Lipschitz classification of surface germs. In
particular, we explain why the outer Lipschitz classification is much harder
than the inner classification, and why the ambient Lipschitz Geometry of
surface germs is very different from their outer Lipschitz Geometry. In
particular, we show that the ambient Lipschitz Geometry of surface germs
includes all of the Knot Theory.
