  The perihelion advance of the orbit of Mercury has long been one of the
observational cornerstones for testing General Relativity (G.R.).
  The main goal of this paper is to discuss how, presently, observational and
theoretical constraints may challenge Einstein's theory of gravitation
characterized by beta=gamma=1. To achieve this purpose, we will first recall
the experimental constraints upon the Eddington-Robertson parameters gamma,
beta and the observational bounds for the perihelion advance of Mercury. A
second point will address the values given, up to now, to the solar quadrupole
moment by several authors. Then, we will briefly comment why we use a recent
theoretical determination of the solar quadrupole moment, J2=(2.0 +/- 0.4)
10^{-7}, which takes into account both surfacic and internal differential
rotation, in order to compute the solar contribution to Mercury's perihelion
advance. Further on, combining bounds on gamma and J2 contributions, and taking
into account the observational data range for the perihelion shift, we will be
able to give a range of values for beta.
  Alternatively, taking into account the observed value of perihelion shift,
one can deduce a dynamical estimation of J2 in the setting of G.R. This point
is important as it provides a solar model independent estimation that can be
confronted with other determinations of J2 based upon solar theory and solar
observations (oscillation data, oblateness...).
  Finally, a glimpse at future satellite experiments will help us to understand
how stronger constraints upon the parameter space (gamma, beta, J2) as well as
a separation of the two contributions (J2, or purely relativistic) might be
expected in the future.
