The chemo-physical parametrisation of stellar spectra is essential for understanding the nature and evolution of stars and of Galactic stellar populations. Gaia DR3 contains the parametrisation of RVS data performed by the General Stellar Parametriser-spectroscopy, module. Here we describe the parametrisation of the first 34 months of RVS observations. GSP-spec estimates the chemo-physical parameters from combined RVS spectra of single stars. The main analysis workflow described here, MatisseGauguin, is based on projection and optimisation methods and provides the stellar atmospheric parameters; the individual chemical abundances of N, Mg, Si, S, Ca, Ti, Cr, FeI, FeII, Ni, Zr, Ce and Nd; the differential equivalent width of a cyanogen line; and the parameters of a DIB feature. Another workflow, based on an artificial neural network, provides a second set of atmospheric parameters that are useful for classification control. We implement a detailed quality flag chain considering different error sources. With about 5.6 million stars, the Gaia DR3 GSP-spec all-sky catalogue is the largest compilation of stellar chemo-physical parameters ever published and the first one from space data. Internal and external biases have been studied taking into account the implemented flags. In some cases, simple calibrations with low degree polynomials are suggested. The homogeneity and quality of the estimated parameters enables chemo-dynamical studies of Galactic stellar populations, interstellar extinction studies from individual spectra, and clear constraints on stellar evolution models. We highly recommend that users adopt the provided quality flags for scientific exploitation . The Gaia DR3 GSP-spec catalogue is a major step in the scientific exploration of Milky Way stellar populations, confirming the Gaia promise of a new Galactic vision (abridged).