  We introduce a metric homotopy theory, which we call Moderately Discontinuous
Homotopy, designed to capture Lipschitz properties of metric singular
subanalytic germs. It matches with the Moderately Discontinuous Homology theory
receantly developed by the authors and E. Sampaio. The $k$-th MD homotopy group
is a group $MDH^b_\bullet$ for any $b\in [1,\infty]$ together with
homomorphisms $MD\pi^b\to MD\pi^{b'}$ for any $b\geq b'$. We develop all its
basic properties including finite presentation of the groups, long homology
sequences of pairs, metric homotopy invariance, Seifert-van Kampen Theorem and
the Hurewicz isomorphism Theorem. We prove comparison theorems that allow to
relate the metric homotopy groups with topological homotopy groups of
associated spaces. For $b=1$ it recovers the homotopy groups of the tangent
cone for the outer metric and of the Gromov tangent cone for the inner one. In
general, for $b=\infty$ the $MD$-homotopy recovers the homotopy of the
punctured germ. Hence, our invariant can be seen as an algebraic invariant
interpolating the homotopy from the germ to its tangent cone. We end the paper
with a full computation of our invariant for any normal surface singularity for
the inner metric. We also provide a full computation of the MD-Homology in the
same case.
