An extensive library of 2500 10 500 Å synthetic spectra

We present a complete library of synthetic spectra based on Kurucz's codes that covers the 2500 10 500 Å wavelength range at resolving powers R_{P}=20 000, 11 500 (≡GAIA), 8500 (≡RAVE), 2000 (≡SLOAN) and uniform dispersions of 1 and 10 Å/pix. The library maps the whole HR diagram, exploring 51 288 combinations of atmospheric parameters spanning the ranges: 3500 ≤ T_eff ≤ 47 500 K, 0.0≤ log g ≤ 5.0 , -2.5 ≤ [M/H] ≤ 0.5, [ α/Fe] = 0.0,+0.4, ξ = 1, 2, 4 km s^-1, 0 ≤ V_rot ≤ 500 km s^-1. The spectra are available both as absolute fluxes as well as continuum normalized. Performance tests and spectroscopic applications of the library are discussed, including automatic classification of data from spectroscopic surveys like RAVE, SLOAN, GAIA. The entire library of synthetic spectra is accessible via the web.