  Carbon nanotubes are a versatile material in which many aspects of condensed
matter physics come together. Recent discoveries, enabled by sophisticated
fabrication, have uncovered new phenomena that completely change our
understanding of transport in these devices, especially the role of the spin
and valley degrees of freedom. This review describes the modern understanding
of transport through nanotube devices.
  Unlike conventional semiconductors, electrons in nanotubes have two angular
momentum quantum numbers, arising from spin and from valley freedom. We focus
on the interplay between the two. In single quantum dots defined in short
lengths of nanotube, the energy levels associated with each degree of freedom,
and the spin-orbit coupling between them, are revealed by Coulomb blockade
spectroscopy. In double quantum dots, the combination of quantum numbers
modifies the selection rules of Pauli blockade. This can be exploited to read
out spin and valley qubits, and to measure the decay of these states through
coupling to nuclear spins and phonons. A second unique property of carbon
nanotubes is that the combination of valley freedom and electron-electron
interactions in one dimension strongly modifies their transport behaviour.
Interaction between electrons inside and outside a quantum dot is manifested in
SU(4) Kondo behavior and level renormalization. Interaction within a dot leads
to Wigner molecules and more complex correlated states.
  This review takes an experimental perspective informed by recent advances in
theory. As well as the well-understood overall picture, we also state clearly
open questions for the field. These advances position nanotubes as a leading
system for the study of spin and valley physics in one dimension where
electronic disorder and hyperfine interaction can both be reduced to a very low
level.
