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International audience; Motivated by the desire to reliably and automatically classify structure of thousands of COSMOS galaxies, we present ZEST, the Zurich Estimator of Structural Types. To classify galaxy structure, ZEST uses (1) five nonparametric diagnostics: asymmetry, concentration, Gini coefficient, second-order moment of the brightest 20% of galaxy pixels, and ellipticity; and (2) the exponent n of single-Sérsic fits to the two-dimensional surface brightness distributions. To fully exploit the wealth of information while reducing the redundancy present in these diagnostics, ZEST performs a principal component (PC) analysis. We use a sample of ~56,000 IAB1, PC2, and PC3. We demonstrate the robustness of the ZEST grid on the z=0 sample of Frei et al. The ZEST classification breaks most of the degeneracy between different galaxy populations that affects morphological classifications based on only some of the diagnostics included in ZEST. As a first application, we present the evolution since z~1 of the luminosity functions (LFs) of COSMOS galaxies of early, disk, and irregular galaxies and, for disk galaxies, of different bulge-to-disk ratios. Overall, we find that the LF up to a redshift z=1 is consistent with a pure luminosity evolution (of about 0.95 mag at z~0.7). We highlight, however, two trends that are in general agreement with a downsizing scenario for galaxy formation, i.e., (1) a deficit of a factor of about 2 at z~0.7 of MB>-20.5 structurally classified early-type galaxies and (2) an excess of a factor of about 3, at a similar redshift, of irregular galaxies. Based on observations with the NASA/ESA Hubble Space Telescope , obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under NASA contract NAS 5-26555 also based on data collected at the Subaru Telescope, which is operated by the National Astronomical Observatory of Japan; Kitt Peak National Observatory, Cerro Tololo Inter-American Observatory, and the National Optical Astronomy Observatory, which are operated by AURA, Inc., under cooperative agreement with the National Science Foundation; and the Canada-France-Hawaii Telescope operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique de France, and the University of Hawaii.