Context : In analyses of stellar spectra and colours , and for the analysis of integrated light from galaxies , a homogeneous grid of model atmospheres of late-type stars and corresponding flux spectra is needed . Aims : To construct an extensive grid of spherically symmetric models ( supplemented with plane-parallel ones for the highest surface gravities ) , built on up-to-date atomic and molecular data , and make it available for public use . Methods : The most recent version of the MARCS program is used . Results : We present a grid of about 10 ^ { 4 } model atmospheres for stars with 2500 { K } \leq T _ { eff } \leq 8000 { K } , -1 \leq log g = log ( GM / R ^ { 2 } ) \leq 5 ( cgs ) with various masses and radii , -5 \leq [ Me/H ] \leq + 1 , with [ \alpha / Fe ] = 0.0 and 0.4 and different choices of C and N abundances , including ” CN-cycled ” models with C/N = 4.07 ( solar ) , 1.5 and 0.5 , C/O ranging from 0.09 to ( normally ) 5.0 to also represent stars of spectral types R , S and N , and with 1.0 \leq \xi _ { t } \leq 5 km/s . We also list thermodynamic quantities ( T , P _ { g } , P _ { e } , \rho , partial pressures of molecules , etc ) and provide them on the World Wide Web , as well as calculated fluxes in approximately 108,000 wavelength points . Underlying assumptions in addition to 1D stratification ( spherical or plane-parallel ) include hydrostatic equilibrium , mixing-length convection and LTE . A number of general properties of the models are discussed , in particular in relation to the effects of changing abundances , of blanketing and of sphericity . We have found and illustrate positive and negative feed-backs between sphericity and molecular blanketing . Models are compared with other available grids and excellent agreement is found with plane-parallel models of Castelli and Kurucz ( if convection is treated consistently ) within the overlapping parameter range . Although there are considerable departures from the spherically symmetric NextGen models , the agreement with more recent PHOENIX models is gratifying . Conclusions : The models of the grid show considerable regularities , but some interesting departures from general patterns occur for the coolest models due to the molecular opacities . We have tested a number of approximate ” rules of thumb ” concerning effects of blanketing and sphericity and found them to often be astonishingly accurate . Some interesting new phenomena have been discovered and explored , 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 line absorption data for molecules , as well as details of models and comparisons with observations in subsequent papers .