  Crystalline topological phases have recently attracted a lot of experimental
and theoretical attention. Key advances include the complete elementary band
representation analyses of crystalline matter by symmetry indicators and the
discovery of higher-order hinge and corner states. However, current
classification schemes of such phases are either implicit or limited in scope.
We present a new scheme for the explicit classification of crystalline
topological insulators and superconductors. These phases are protected by
crystallographic point group symmetries and are characterized by bulk
topological invariants. The classification paradigm generalizes the Clifford
algebra extension process of each Altland-Zirnbauer symmetry class and utilizes
algebras which incorporate the point group symmetry. Explicit results for all
point group symmetries of three-dimensional crystals are presented as well as
for all symmorphic layer groups of two-dimensional crystals. We discuss future
extensions for treatment of magnetic crystals and defected or
higher-dimensional systems as well as weak and fragile invariants.
