The properties of HgTe/Hg1−x Cdx Te superlattices (SLs) and quantum wells, (QWs), are of fundamental interest; SLs are potentially useful for infrared opto-electronic applications and QWs have aroused much interest due to recent evidence that they are topological insulators with dissipation less transport in edge states in the absence of a magnetic field. Depending on the HgTe thickness , the band structure has either the normal sequence of quantum well states or an inverted one when nm, in which the electron E1 subband is a valence subband and lies below the heavy hole H1 subband. This has been shown to lead to a very large Rashba spin–orbit (s–o) splitting of up to meV in HgTe/Hg1−x Cdx Te QWs with an inverted band structure. This as well as other transport results will be presented. These properties of the HgTe/Hg1−x Cdx Te QW, large s–o splitting and an inverted band structure, have recently been shown to result in the quantum spin Hall effect. In the quantum spin Hall effect, carriers with opposite spin move in opposite directions on a given edge in an insulator in the absence of a magnetic field. Convincing evidence for non-local edge channel transport in HgTe based QWs has been recently published. The results demonstrate that quantum transport through helical edge channels is dissipation less at zero magnetic field and agree quantitatively with the theory of the quantum spin Hall effect. Images of current in the edge channels by means of scanning gate microscopy have revealed well localized scattering sites which perturb the quantum spin Hall edge states on a sub-micrometer scale. In the micrometer sized regions between the scattering sites, current appeared to propagate unperturbed Recently, images of edge channels in HgTe QWs in the quantum spin Hall regime have been obtained by means of their magnetic fields with a scanning superconducting quantum interference device (SQUID). The edge channels were observed to dominate transport in devices with edges much longer than the scattering length, and to persist even in the presence of bulk conduction.