  Different experiments are ongoing to measure the effect of gravity on cold
neutral antimatter atoms such as positronium, muonium and antihydrogen. Among
those, the project GBAR in CERN aims to measure precisely the gravitational
fall of ultracold antihydrogen atoms. In the ultracold regime, the interaction
of antihydrogen atoms with a surface is governed by the phenomenon of quantum
reflection which results in bouncing of antihydrogen atoms on matter surfaces.
This allows the application of a filtering scheme to increase the precision of
the free fall measurement. In the ultimate limit of smallest vertical
velocities, antihydrogen atoms are settled in gravitational quantum states in
close analogy to ultracold neutrons (UCNs). Positronium is another neutral
system involving antimatter for which free fall under gravity is currently
being investigated at UCL. Building on the experimental techniques under
development for the free fall measurement, gravitational quantum states could
also be observed in positronium. In this contribution, we review the status of
the ongoing experiments and discuss the prospects of observing gravitational
quantum states of antimatter and their implications.
