[en] After a review of some theoretical formulations of the concept of total reaction cross σR, involving low energy (∼ < 10 MeV/amu) and high energy (> GeV/amu) models, two experimental methods used for direct measurements of σR, are presented, namely the beam attenuation and the associated γ-rays 4π detection methods. Then a number of experimental results of σR in the Fermi energy range is given and the data are compared on the one hand with the predictions of the low energy BASS model (assuming a classical one-dimensional nucleus-nucleus potential interaction), and on the other hand with the predictions of a ''high-energy'' microscopic calculation performed using the formalism of KAROL (assuming that nuclear reactions are produced by individual nucleon-nucleon interactions). Finally, some experimental and theoretical problems are discussed and general conclusions are tentatively proposed. 43 refs