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© 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society. Thanks to R. Sheth for comments on an early draft. Thanks to the anonymous referee for a constructive and quick report., We quantify the evolution of the stellar mass functions (SMFs) of star-forming and quiescent galaxies as a function of morphology from z ∼ 3 to the present. Our sample consists of ∼50 000 galaxies in the CANDELS fields (∼880 arcmin^2), which we divide into four main morphological types, i.e. pure bulge-dominated systems, pure spiral disc-dominated, intermediate two-component bulge+disc systems and irregular disturbed galaxies. At z ∼ 2, 80 per cent of the stellar mass density of star-forming galaxies is in irregular systems. However, by z ∼ 0.5, irregular objects only dominate at stellar masses below 10^9 M_⊙. A majority of the star-forming irregulars present at z ∼ 2 undergo a gradual transformation from disturbed to normal spiral disc morphologies by z ∼ 1 without significant interruption to their star formation. Rejuvenation after a quenching event does not seem to be common except perhaps for the most massive objects, because the fraction of bulge-dominated star-forming galaxies with M^*/M_⊙ > 10^10.7 reaches 40 per cent at z < 1. Quenching implies the presence of a bulge: the abundance of massive red discs is negligible at all redshifts over 2 dex in stellar mass. However, the dominant quenching mechanism evolves. At z > 2, the SMF of quiescent galaxies above M^* is dominated by compact spheroids. Quenching at this early epoch destroys the disc and produces a compact remnant unless the star-forming progenitors at even higher redshifts are significantly more dense. At 1 < z < 2, the majority of newly quenched galaxies are discs with a significant central bulge. This suggests that mass quenching at this epoch starts from the inner parts and preserves the disc. At z < 1, the high-mass end of the passive SMF is globally in place and the evolution mostly happens at stellar masses below 10^10 M_⊙. These low-mass galaxies are compact, bulge-dominated systems, which were environmentally quenched: destruction of the disc through ram-pressure stripping is the likely proc, Depto. de Física de la Tierra y Astrofísica, Fac. de Ciencias Físicas, TRUE, pub