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

Lipschitz Geometry of Real Semialgebraic Surfaces