  A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a
$C^1$-neighbourhood $\cU$ of $f$ such that every diffeomorphism in $g\in \cU$
has a hyperbolic set $\La_g$, depending continuously on $g$, such that the
stable and unstable manifolds of $\La_g$ have some non-transverse intersection.
For every manifold of dimension greater than or equal to three, we exhibit a
local mechanism (blender-horseshoes) generating diffeomorphisms with
$C^1$-robust homoclinic tangencies.
  Using blender-horseshoes, we prove that homoclinic classes of $C^1$-generic
diffeomorphisms containing saddles with different indices and that do not admit
dominated splittings (of appropriate dimensions) display $C^1$-robust
homoclinic tangencies.
