  We present a complete classification of all 1D and 2D orbifold
compactifications. There exist 2 one-dimensional and 17 two-dimensional
orbifolds. The classification includes orbifolds such as S^1/Z_2 or T^2/Z_n, as
well as less familiar ones like T^2/D_n or the Mobius strip. We derive the
explicit form of the basis functions and prove their orthonormality and
completeness. Our study is based on the classification of space groups, which
is well-known from crystallography. We define these groups in a novel, purely
algebraic way. That enables us to determine all possible parities that can be
defined on the orbifolds. We discuss field theories on T^2/Z_n with brane
kinetic terms, and describe the derivation of their mass eigenstate bases.
