  We formulate topological crystalline materials on the basis of the twisted
equivariant $K$-theory. Basic ideas of the twisted equivariant $K$-theory is
explained with application to topological phases protected by crystalline
symmetries in mind, and systematic methods of topological classification for
crystalline materials are presented. Our formulation is applicable to bulk
gapful topological crystalline insulators/superconductors and their gapless
boundary and defect states, as well as bulk gapless topological materials such
as Weyl and Dirac semimetals, and nodal superconductors. As an application of
our formulation, we present a complete classification of topological
crystalline surface states, in the absence of time-reversal invariance. The
classification works for gapless surface states of three-dimensional
insulators, as well as full gapped two-dimensional insulators. Such surface
states and two-dimensional insulators are classified in a unified way by 17
wallpaper groups, together with the presence or the absence of (sublattice)
chiral symmetry. We identify the topological numbers and their representations
under the wallpaper group operation. We also exemplify the usefulness of our
formulation in the classification of bulk gapless phases. We present a new
class of Weyl semimetals and Weyl superconductors that are topologically
protected by inversion symmetry.
