Well after the discovery of the global acoustic oscillations of the Sun, (p-modes), it was realized that another type of mode should exist in such a stratified medium: the internal gravity modes. It was also noticed that this type of mode was only providing information on the inner 40% of the Sun; precisely the region where p-modes are less sensitive to the physical conditions. This fact turned the g-modes into the most powerful tool for the investigation of the solar core.Excepting the 160m oscillation/artifact /16/, the first observational claim of internal g-mode detection took place in 1983 /14/, and, since then, a lot of effort has been devoted to this subject both, observationally and in data analysis, and in the theoretical field. In the last seven years, three different principal data sets have been extensively analized in a search for g-modes (Tenerife, ACRIM and Stanford). Their characteristics and the results claimed from the analysis will be reviewed and discussed. However the present situation is that the discrepancies between observers suggest that the solar g-modes though probably detected, are not yet measured nor classified. What is of general agreement is the existence of signals, embedded in noise, in the low frequency range where g-modes are expected to lie. As we will emphasize in this review, we are dealing with a far different situation than for the p-modes: lower frequencies, poorer signal-to-noise ratio, etc... Taking into account these difficulties, there is a real hope of success in the near future: with the space experiments devoted to Helioseismology on board of SOHO and the earth-based networks of helioseismic instruments.