In the 1950's interest arose in the science-fiction idea of "antigravity". This concept was a speculation that matter and antimatter would repel each other in ordinary tensor (Einsteinian) gravity. (This was reminiscent of the Coulomb interaction between like charges.) Three types of arguments were raised to disprove "antigravity": (i) violation of conservation of energy. (ii) violation of the principle of equivalence, (iii) anomalous regeneration of K(S) mesons, all of which were unseen. Unfortunately, in the physics folklore these arguments came to be interpreted as ruling out any difference in the gravitational interaction of matter and antimatter to the earth, not just ruling out this "antigravity." After reviewing the history behind the concept of antigravity, we discuss modern theoretical ideas and experiments which are relevant to the possibility that there exist static, non-Newtonian components of gravity. We then consider the arguments against antigravity, pointing out their regimes of applicability and inapplicability to modern quantum theories of gravity. We specifically show that, if these arguments are applied to the ongoing experiment to measure the gravitational acceleration of the antiproton, they do not rule out a large anomalous gravitational response for the antiproton. The antiproton gravity experiment is one of five types of experiments that we consider to be the most critical in testing for finite but long-range, non-Newtonian and non-Einsteinian gravity. We discuss how all five are related to each other and how their completion would improve our knowledge of gravitational forces.