  The Van Hove singularity (VHS) provides a paradigm for the study of the role
of peaks in the density of states (dos) on electronic properties. More
importantly, it appears to play a major role in the physics of the high-T_c
superconductors, particularly since recent photoemission studies have found
that the VHS is close to the Fermi level in most of the high-T_c cuprates near
the composition of optimum T_c. This paper offers a comprehensive survey of the
VHS model, describing both theoretical properties and experimental evidence for
the picture. Special topics discussed include a survey of the Fermi surfaces of
the cuprates and related compounds, and an analysis of the reliability of the
slave boson approach to correlation effects.
  While many properties of the cuprates can be qualitatively understood by a
simple rigid-band-filling model, this is inadequate for more quantitative
results, since correlation effects tend to pin the Fermi level near the VHS
over an extended doping range, and can lead to a nanoscale phase separation.
Furthermore, the peaks in the dos lead to competition from other instabilities,
both magnetic and structural (related to charge density waves). A novel form of
dynamic structural instability, involving dynamic VHS-Jahn-Teller effects has
been predicted. Scattered through the literature, there is considerable
experimental evidence for both nanoscale phase separation of holes, and for
local, possibly dynamic, structural disorder. This review attempts to gather
these results into a comprehensive database, to sort the results, and to see
how they fit into the Van Hove scenario. Recent experiments on underdoped
cuprates are found to provide a strong confirmation that the pseudogap is
driven by a splitting of the VHS degeneracy.
