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We have constructed a grid of about 10,000 spherically symmetric and plane-parallel models with the MARCS program, and make it available for public use. Parameter ranges are: Teff=2500 to 8000 K, log g =log(GM/R2)= -1 to 5 (cgs) with various masses and radii, [Me/H]=-5 to +1, with [Alpha/Fe] = 0.0 and 0.4 and different choices of C and N abundances to also represent stars of types R, S and N, and with microturbulence parameters from 1 to 5 km/s. We also list fluxes in approximately 108,000 wavelength points. Underlying assumptions in addition to 1D stratification include hydrostatic equilibrium, MLT convection and LTE. A number of general properties of the models are discussed, in relation to the effects of changing blanketing and sphericity. Models are compared with other available grids and excellent agreement is found with plane-parallel models of Castelli and Kurucz within the overlapping parameter range. Although there are departures from the spherically symmetric NextGen models, the agreement with more recent PHOENIX models is gratifying. The models of the grid show regularities, but some interesting departures from general patterns occur for the coolest models due to the molecular opacities. We have tested rules of thumb concerning effects of blanketing and sphericity and found them to often be astonishingly accurate. Some interesting new phenomena have been discovered, such as the intricate coupling between blanketing and sphericity, and the strong effects of carbon enhancement on metal-poor models. We give further details of models and comparisons with observations in subsequent papers.