  We have performed a computational screening of topological two-dimensional
(2D) materials from the Computational 2D Materials Database (C2DB) employing
density functional theory. A full \textit{ab initio} scheme for calculating
hybrid Wannier functions directly from the Kohn-Sham orbitals has been
implemented and the method was used to extract $\mathbb{Z}_2$ indices, Chern
numbers and Mirror Chern numbers of 3331 2D systems including both
experimentally known and hypothetical 2D materials. We have found a total of 46
quantum spin Hall insulators, 7 quantum anomalous Hall insulators and 9
crystalline topological insulators that are all predicted to be dynamically
stable. Roughly one third of these were known prior to the screening. The most
interesting of the novel topological insulators are investigated in more
detail. We show that the calculated topological indices of the quantum
anomalous Hall insulators are highly sensitive to the approximation used for
the exchange-correlation functional and reliable predictions of the topological
properties of these materials thus require methods beyond density functional
theory. We also performed $GW$ calculations, which yield a gap of 0.65 eV for
the quantum spin Hall insulator PdSe$_2$ in the MoS$_2$ crystal structure. This
is significantly higher than any known 2D topological insulator and three times
larger than the Kohn-Sham gap.
