  Mindful of the anomalous perihelion precession of Mercury discovered by U. Le
Verrier in the second half of the nineteenth century and its successful
explanation by A. Einstein with his General Theory of Relativity in the early
years of the twentieth century, discrepancies among observed effects in our
Solar system and their theoretical predictions on the basis of the currently
accepted laws of gravitation applied to known bodies have the potential of
paving the way for remarkable advances in fundamental physics. This is
particularly important now more than ever, given that most of the Universe
seems to be made of unknown substances dubbed Dark Matter and Dark Energy.
Should this not be directly the case, Solar system's anomalies could anyhow
lead to advancements in cumulative science, as shown to us by the discovery of
Neptune in the first half of the nineteenth century. Moreover, investigations
in one of such directions can serendipitously enrich the other one as well. The
current status of some alleged gravitational anomalies in the Solar system is
critically reviewed. They are: a) Possible anomalous advances of planetary
perihelia; b) Unexplained orbital residuals of a recently discovered moon of
Uranus (Mab); c) The lingering unexplained secular increase of the eccentricity
of the orbit of the Moon; d) The so-called Faint Young Sun Paradox; e) The
secular decrease of the mass parameter of the Sun; f) The Flyby Anomaly; g) The
Pioneer Anomaly; and h) The anomalous secular increase of the astronomical unit
