  Tests of the Weak Equivalence Principle (WEP) probe the foundations of
physics. Ever since Galileo in the early 1600s, WEP tests have attracted some
of the best experimentalists of any time. Progress has come in bursts, each
stimulated by the introduction of a new technique: the torsion balance, signal
modulation by Earth rotation, the rotating torsion balance. Tests for various
materials in the field of the Earth and the Sun have found no violation to the
level of about 1 part in 1e13. A different technique, Lunar Laser Ranging
(LLR), has reached comparable precision. Today, both laboratory tests and LLR
have reached a point when improving by a factor of 10 is extremely hard. The
promise of another quantum leap in precision rests on experiments performed in
low Earth orbit. The Microscope satellite, launched in April 2016 and currently
taking data, aims to test WEP in the field of Earth to 1e-15, a 100-fold
improvement possible thanks to a driving signal in orbit almost 500 times
stronger than for torsion balances on ground. The `Galileo Galilei' (GG)
experiment, by combining the advantages of space with those of the rotating
torsion balance, aims at a WEP test 100 times more precise than Microscope, to
1e-17. A quantitative comparison of the key issues in the two experiments is
presented, along with recent experimental measurements relevant for GG. Early
results from Microscope, reported at a conference in March 2017, show
measurement performance close to the expectations and confirm the key role of
rotation with the advantage (unique to space) of rotating the whole spacecraft.
Any non-null result from Microscope would be a major discovery and call for
urgent confirmation; with 100 times better precision GG could settle the matter
and provide a deeper probe of the foundations of physics.
