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Recently, new solar model atmospheres have been developed to replace classical 1D LTE hydrostatic models and used to for example derive the solar chemical composition. We aim to test various models against key observational constraints. In particular, a 3D model used to derive the solar abundances, a 3D MHD model (with an imposed 10 mT vertical magnetic field), 1D models from the PHOENIX project, the 1D MARCS model, and the 1D semi-empirical model of Holweger & M\"uller. We confront the models with observational diagnostics of the temperature profile: continuum centre-to-limb variations (CLV), absolute continuum fluxes, and the wings of hydrogen lines. We also test the 3D models for the intensity distribution of the granulation and spectral line shapes. The predictions from the 3D model are in excellent agreement with the continuum CLV observations, performing even better than the Holweger & M\"uller model (constructed largely to fulfil such observations). The predictions of the 1D theoretical models are worse, given their steeper temperature gradients. For the continuum fluxes, predictions for most models agree well with the observations. No model fits all hydrogen lines perfectly, but again the 3D model comes ahead. The 3D model also reproduces the observed continuum intensity fluctuations and spectral line shapes very well. The excellent agreement of the 3D model with the observables reinforces the view that its temperature structure is realistic. It outperforms the MHD simulation in all diagnostics, implying that recent claims for revised abundances based on MHD modelling are premature. Several weaknesses in the 1D models are exposed. The differences between the PHOENIX LTE and NLTE models are small. We conclude that the 3D hydrodynamical model is superior to any of the tested 1D models, which gives further confidence in the solar abundance analyses based on it., Comment: 17 pages, 15 figures. Accepted for publication in A&A