  We present a scheme to explicitly construct and classify general topological
states jointly protected by an onsite symmetry group and a spatial symmetry
group. We show that all these symmetry protected topological states can be
adiabatically deformed (allowing for stacking of trivial states) into a special
class of states we call topological crystals. A topological crystal in, for
example, three dimensions is a real-space assembly of finite-sized pieces of
topological states in one and two dimensions protected by the local symmetry
group alone, arranged in a configuration invariant under the spatial group and
glued together such there is no open edge or end. As a demonstration of
principle, we explicitly enumerate all inequivalent topological crystals for
non-interacting time-reversal symmetric electronic insulators with significant
spin-orbit coupling and any one of the 230 space groups in three dimensions.
Because every topological crystalline insulator can be deformed into a
topological crystal, the enumeration of the latter gives topological
crystalline insulators a full classification and for each class an explicit
real-space construction. We also extend these results to give a unified
classification including both strong topological insulators and topological
crystalline insulators.
