  We introduce a new metric homology theory, Moderately Discontinuous Homology,
which captures Lipschitz properties of metric subanalytic germs. The main
novelty is to allow "moderately discontinuous" chains, which are specially
advantageous for capturing the subtleties of the outer metric phenomena. Our
invariant is a finitely generated graded abelian group $MDH^b_\bullet$ for any
$b\in [1,\infty]$ and homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ for any
$b\geq b'$. Here $b$ is a "discontinuity rate". The homology groups for the
inner or outer metric are proved to be finitely generated and that only
finitely many homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ are essential.
For $b=1$ it recovers the homology of the tangent cone for the outer metric and
of the Gromov tangent cone for the inner one. In general, for $b=\infty$ the
$MD$- homology recovers the homology of the punctured germ. Hence, our
invariant interpolates from the germ to its tangent cone. Our homology theory
is a bi-Lipschitz subanalitic invariant, is invariant by suitable metric
homotopies, and satisfies versions of the relative and Mayer-Vietoris long
exact sequences. Moreover, fixed a discontinuity rate $b$ we show that it is
functorial for a class of discontinuous Lipschitz maps, whose discontinuities
are $b$-moderated; this makes the theory quite flexible. In the complex
analytic setting we introduce an enhancement called Framed MD Homology, which
takes into account information from fundamental classes. As applications we
prove that Moderately Discontinuous Homology characterizes smooth germs among
all complex analytic germs, recovers the number of irreducible components of
complex analytic germs and the embedded topological type of plane branches.
Framed MD Homology recovers the topological type of any plane curve singularity
and relative multiplicities of complex analytic germs.
