Spectrophotometric observations for 88 red supergiant candidates in the Large Magellanic Cloud are presented. The spectra range from 4800 to 7700Angstroms with a resolution of 10Angstroms. The error in the absolute fluxes is 0.04 to 0.05mag. The molecular bands of the member stars are often rather weak, i.e. many of these are not M- but K-type supergiants. The data are available on the Strasbourg stellar data base (CDS). Most of the red (super)giant model atmospheres available up to now do not reproduce the observations well. The models of Kurucz and Lejeune, Cuisinier & Buser - often applied especially to population synthesis - correctly describe the strengths of atomic lines and the overall increase of the flux towards the red, but strongly underestimate the strengths of molecular bands. The models presented by Plez, however, tend to reproduce the observed spectra well, except for the blue, as they include a more complete list of opacity sources. Concerning physical properties, only the Plez models give reliable results. Considering the relation between effective temperature and the strengths of molecular bands, both the Kurucz and Lejeune models predict much higher temperatures than derived from the interferometric radius measurements discussed by Schmidt-Kaler and Dyck et al. The temperatures given by the Plez models show a much better agreement with these observations. Furthermore, the relation between T_eff and molecular absorption is much more clearly defined. When considering metallicities, however, the Plez models also fail, as they predict a [Fe/H] distribution that is much too broad, and furthermore an increase of T_eff with increasing [Fe/H] which clearly contradicts models of stellar evolution. The effective temperatures based on the Plez models range mostly from 3500 to 4100K. The surface gravities derived on the basis of the Geneva evolutionary models range from logg=-0.3 to 0.3, while the bolometric luminosities based on BVRIJHK observations range from -6.6 to -9.0mag. Except for the most luminous objects, the stars form a well-defined strip in the Hertzsprung-Russell diagram extending to the Hayashi limit at 3500K. At present, reliable metallicities cannot be derived because of inadequacies in the model atmospheres.