  In this article, we review the recent progress in the study of topological
phases in systems with space-time inversion symmetry $I_{\text{ST}}$.
$I_{\text{ST}}$ is an anti-unitary symmetry which is local in momentum space
and satisfies $I_{\text{ST}}^2=1$ such as $PT$ or $C_{2}T$ symmetry where $P$,
$T$, $C_2$ indicate inversion, time-reversal, and two-fold rotation symmetries,
respectively. Under $I_{\text{ST}}$, the Hamiltonian and the Bloch wave
function can be constrained to be real-valued, which makes the Berry curvature
and the Chern number to vanish. In this class of systems, gapped band
structures of real wave functions can be topologically distinguished by
Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers
$w_1$ and $w_2$, respectively, are the corresponding invariants in 1D and 2D,
which are equivalent to the quantized Berry phase and the $Z_2$ monopole
charge, respectively. We first describe the topological phases characterized by
the first Stiefel-Whitney number, including 1D topological insulators with
quantized charge polarization, 2D Dirac semimetals, and 3D nodal line
semimetals. Next we review how the second Stiefel-Whitney class characterizes
the 3D nodal line semimetals carrying a $Z_{2}$ monopole charge. In particular,
we explain how the second Stiefel-Whitney number $w_2$, the $Z_{2}$ monopole
charge, and the linking number between nodal lines are related. Finally, we
review the properties of 2D and 3D topological insulators characterized by the
nontrivial second Stiefel Whitney class.
