  This paper reviews and summarizes the recent first-principles theoretical
studies of the structural stability, electronic structure, optical and magnetic
properties of nonmetal-doped TiO$_2$. The first section presents a comparison
study of the structural stability for X-anion and X-cation doped TiO$_2$ (X=B,
C, Si, Ge, N, P, As, Sb, S, Se, Te, F, Cl, Br, and I), which reveals that the
sites of nonmetal dopants (i.e., at O sites or at Ti sites) in TiO$_2$ are
determined by the growth condition of doped TiO$_2$ and the dopants'
electronegativities. The next section reviews the electronic structure, optical
absorption and mechanism of the visible-light photocatalytic activity for
nonmetal-doped TiO$_2$. The third section summarizes the origin of the
spin-polarization and the magnetic coupling character in C- (N- and B-) doped
TiO$_2$.
