  One-dimensional helical liquids can appear at boundaries of certain condensed
matter systems. Two prime examples are the edge of a quantum spin Hall
insulator and the hinge of a three-dimensional second-order topological
insulator. For these materials, the presence of a helical state at the boundary
serves as a signature of their nontrivial electronic bulk topology.
Additionally, these boundary states are of interest themselves, as a novel
class of strongly correlated low-dimensional systems with interesting potential
applications. Here, we review existing results on such helical liquids in
semiconductors. Our focus is on the theory, though we confront it with existing
experiments. We discuss various aspects of the helical liquids, such as their
realization, topological protection and stability, or possible experimental
characterization. We lay emphasis on the hallmark of these states, being the
prediction of a quantized electrical conductance. Since so far reaching a
well-quantized conductance has remained challenging experimentally, a large
part of the review is a discussion of various backscattering mechanisms which
have been invoked to explain this discrepancy. Finally, we include topics
related to proximity-induced topological superconductivity in helical states,
as an exciting application towards topological quantum computation with the
resulting Majorana bound states.
