  We survey recent experimental results including quantum oscillations and
complementary measurements probing the electronic structure of underdoped
cuprates, and theoretical proposals to explain them. We discuss quantum
oscillations measured at high magnetic fields in the underdoped cuprates that
reveal a small Fermi surface section comprising quasiparticles that obey
Fermi-Dirac statistics, unaccompanied by other states of comparable
thermodynamic mass at the Fermi level. The location of the observed Fermi
surface section at the nodes is indicated by a body of evidence including the
collapse in Fermi velocity measured by quantum oscillations, which is found to
be associated with the nodal density of states observed in angular resolved
photoemission, the persistence of quantum oscillations down to low fields in
the vortex state, the small value of density of states from heat capacity and
the multiple frequency quantum oscillation pattern consistent with nodal
magnetic breakdown of bilayer-split pockets. A nodal Fermi surface pocket is
further consistent with the observation of a density of states at the Fermi
level concentrated at the nodes in photoemission experiments, and the antinodal
pseudogap observed by photoemission, optical conductivity, nuclear magnetic
resonance Knight shift, as well as other complementary diffraction, transport
and thermodynamic measurements. One of the possibilities considered is that the
small Fermi surface pockets observed at high magnetic fields can be understood
in terms of Fermi surface reconstruction by a form of small wavevector charge
order, observed over long lengthscales in experiments such as nuclear magnetic
resonance and x-ray scattering, potentially accompanied by an additional
mechanism to gap the antinodal density of states.
