  We review the Lieb-Schultz-Mattis theorem and its variants, which are no-go
theorems that state that a quantum many-body system with certain conditions
cannot have a locally-unique gapped ground state. We restrict ourselves to
one-dimensional quantum spin systems and discuss both the generalized
Lieb-Schultz-Mattis theorem for models with U(1) symmetry and the extended
Lieb-Schultz-Mattis theorem for models with discrete symmetry. We also discuss
the implication of the same arguments to systems on the infinite cylinder, both
with the periodic boundary conditions and with the spiral boundary conditions.
  For models with U(1) symmetry, we here present a rearranged version of the
original proof of Lieb, Schultz, and Mattis based on the twist operator. As the
title suggests we take a modern topological point of view and prove the
generalized Lieb-Schultz-Mattis theorem by making use of a topological index
(which coincides with the filling factor). By a topological index, we mean an
index that characterizes a locally-unique gapped ground state and is invariant
under continuous (or smooth) modification of the ground state.
  For models with discrete symmetry, we describe the basic idea of the most
general proof based on the topological index introduced in the context of
symmetry-protected topological phases. We start from background materials such
as the classification of projective representations of the symmetry group.
  We also review the notion that we call a locally-unique gapped ground state
of a quantum spin system on an infinite lattice and present basic theorems.
This notion turns out to be natural and useful from the physicists' point of
view.
  We have tried to make the present article readable and almost self-contained.
We only assume basic knowledge about quantum spin systems.
