  Using a systematic relation between topological gapless phases in three
dimensions and topological gapped phases in two dimensions, we identify four
types of higher-order topological semimetals or nodal superconductors (HOTS),
hosting (i) flat zero-energy "Fermi arcs" at crystal hinges, (ii) flat
zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or
helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only
at finite momentum. Bulk-boundary correspondence relates the hinge states to
the bulk topology protecting the nodal point or loop. We classify all HOTS for
all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as
mirror, twofold rotation, or inversion.
