  We review the main arguments against antigravity, a different acceleration of
antimatter relative to matter in a gravitational field, discussing and
challenging Morrison's, Good's and Schiff's arguments. Following Price, we show
that, very surprisingly, the usual expression of the Equivalence Principle is
violated by General Relativity when particles of negative mass are supposed to
exist, which may provide a fundamental explanation of MOND phenomenology,
obviating the need for Dark Matter. Motivated by the observation of repulsive
gravity under the form of Dark Energy, and by the fact that our universe looks
very similar to a coasting (neither decelerating nor accelerating) universe, we
study the Dirac-Milne cosmology, a symmetric matter-antimatter cosmology where
antiparticles have the same gravitational properties as holes in a
semiconductor. Noting the similarities with our universe (age, SN1a luminosity
distance, nucleosynthesis, CMB angular scale), we focus our attention on
structure formation mechanisms, finding strong similarities with our universe.
Additional tests of the Dirac-Milne cosmology are briefly reviewed, and we
finally note that a crucial test of the Dirac-Milne cosmology will be soon
realized at CERN next to the ELENA antiproton decelerator, possibly as early as
fall 2018, with the AEgIS, ALPHA-g and Gbar antihydrogen gravity experiments.
