Does the non-minimal coupling of the scalar field improve or destroy inflation ?

The non-minimal coupling of a scalar field to the Ricci curvature in a curved spacetime is unavoidable according to several authors. The coupling constant is not a free parameter: the prescriptions for the value of the coupling constant in specific scalar field and gravity theories (in particular in general relativity) are studied. The results are applied to the most popular inflationary scenarios of cosmology and their theoretical consistence is analysed. Certain observational constraints on the coupling constant are also discussed.