We present the galaxy stellar mass function (GSMF) to redshift z 1, based on the analysis of about 8500 galaxies with I < 22.5 (AB mag) over 1.4 deg 2 , which are part of the zCOSMOS-bright 10k spectroscopic sample. We investigate the total GSMF, as well as the contributions of early-and late-type galaxies (ETGs and LTGs, respectively), defined by different criteria (broad-band spectral energy distribution, morphology, spectral properties, or star formation activities). We unveil a galaxy bimodality in the global GSMF, whose shape is more accurately represented by 2 Schechter functions, one linked to the ETG and the other to the LTG populations. For the global population, we confirm a mass-dependent evolution ("mass-assembly downsizing"), i.e., galaxy number density increases with cosmic time by a factor of two between z = 1 and z = 0 for intermediate-to-low mass (log(M/M ) ∼ 10.5) galaxies but less than 15% for log(M/M ) > 11. We find that the GSMF evolution at intermediateto-low values of M (log(M/M ) < 10.6) is mostly explained by the growth in stellar mass driven by smoothly decreasing star formation activities, despite the redder colours predicted in particular at low redshift. The low residual evolution is consistent, on average, with ∼0.16 merger per galaxy per Gyr (of which fewer than 0.1 are major), with a hint of a decrease with cosmic time but not a clear dependence on the mass. From the analysis of different galaxy types, we find that ETGs, regardless of the classification method, increase in number density with cosmic time more rapidly with decreasing M, i.e., follow a top-down building history, with a median "building redshift" increasing with mass (z > 1 for log(M/M ) > 11), in contrast to hierarchical model predictions. For LTGs, we find that the number density of blue or spiral galaxies with log(M/M ) > 10 remains almost constant with cosmic time from z ∼ 1. Instead, the most extreme population of star-forming galaxies (with high specific star formation), at intermediate/high-mass, rapidly decreases in number density with cosmic time. Our data can be interpreted as a combination of different effects. Firstly, we suggest a transformation, driven mainly by SFH, from blue, active, spiral galaxies of intermediate mass to blue quiescent and subsequently (1−2 Gyr after) red, passive types of low specific star formation. We find an indication that the complete morphological transformation, probably driven by dynamical processes, into red spheroidal galaxies, occurred on longer timescales or followed after 1−2 Gyr. A continuous replacement of blue galaxies is expected to be accomplished by low-mass active spirals increasing their stellar mass. We estimate the growth rate in number and mass density of the red galaxies at different redshifts and masses. The corresponding fraction of blue galaxies that, at any given time, is transforming into red galaxies per Gyr, due to the quenching of their SFR, is on average ∼25% for log(M/M ) < 11. We conclude that the build-up of galaxies and in particular of...