  The skin effect, which is unique to non-Hermitian systems, can generate an
extensive number of eigenstates localized near the boundary in an open
geometry. Here we propose that in 2D and 3D other quantities besides charge
density are susceptible to the skin effect. We show that 2D and 3D models that
are a hybrid between topological insulators and skin-effect systems can have a
topological skin effect where an extensive number of topological modes, and the
corresponding bulk topological invariant, are pinned to the surface. A key
example, which we call the quantum skin Hall effect is constructed from layers
of Chern insulators and exhibits an extensive Hall conductance and number of
chiral modes bound to surfaces normal to the stack of layers. The same
procedure is further extended to other symmetry classes to illustrate that a
variety of 1D and 2D topological invariants ($\mathbb{Z}$ or $\mathbb{Z}_2)$
are subject to the skin effect. Indeed, we also propose a hybrid 2D system that
exhibits an extensive number of topological corner modes and may be more easily
realized in meta-material experiments.
