  Topological crystalline insulators define a new class of topological
insulator phases with gapless surface states protected by crystalline
symmetries. In this work, we present a general theory to classify topological
crystalline insulator phases based on the representation theory of space
groups. Our approach is to directly identify possible nontrivial surface states
in a semi-infinite system with a specific surface, of which the symmetry
property can be described by 17 two-dimensional space groups. We reproduce the
existing results of topological crystalline insulators, such as mirror Chern
insulators in the $pm$ or $pmm$ groups, $C_{nv}$ topological insulators in the
$p4m$, $p31m$ and $p6m$ groups, and topological nonsymmorphic crystalline
insulators in the $pg$ and $pmg$ groups. Aside from these existing results, we
also obtain the following new results: (1) there are two integer mirror Chern
numbers ($\mathbb{Z}^2$) in the $pm$ group but only one ($\mathbb{Z}$) in the
$cm$ or $p3m1$ group for both the spinless and spinful cases; (2) for the $pmm$
($cmm$) groups, there is no topological classification in the spinless case but
$\mathbb{Z}^4$ ($\mathbb{Z}^2$) classifications in the spinful case; (3) we
show how topological crystalline insulator phase in the $pg$ group is related
to that in the $pm$ group; (4) we identify topological classification of the
$p4m$, $p31m$, and $p6m$ for the spinful case; (5) we find topological
non-symmorphic crystalline insulators also existing in $pgg$ and $p4g$ groups,
which exhibit new features compared to those in $pg$ and $pmg$ groups. We
emphasize the importance of the irreducible representations for the states at
some specific high-symmetry momenta in the classification of topological
crystalline phases. Our theory can serve as a guide for the search of
topological crystalline insulator phases in realistic materials.
