  A summary is given of the high-lights during the Reykjavik Midsummer
Symposium on Non-Linear Phenomena in Accretion Discs around Black Holes. Such
high-lights include the recent advances on understanding: 1) the accretion disc
solution branch dominated by advection (i.e., advection dominated accretion
flows, ADAFs), 2) the importance of magnetic fields in many different respects,
most importantly being responsible for the self-sustained MHD-turbulence giving
rise to the disc viscosity, and 3) the details of the radiation processes
giving rise to the X/gamma-ray continuum originating close to the black hole.
Some old advances are unfortunately also necessary to discuss here. It is
pointed out to the accretion disc research community, that many of the research
papers published on ADAFs 1994-1997 do not accurately present the history of
ADAF research. Some of the results that were presented as new and original
actually appeared in a paper by Ichimaru already in 1977. Also not quoted are
the papers from the 1970's and 1980's calculating the temperature structure of
and the spectra from quasi-spherical accretion flows onto black holes. As the
ADAFs are close to being quasi-spherical, the resulting spectra of ADAFs and
quasi-spherical flows are almost identical. Further of the recent ADAF results
are therefore not new results as is sometimes claimed. In spite of all the
recent progress of various aspects of accretion flows around black holes, many
of the research lines have still not been merged providing potential for
further dramatic progress in coming years.
