A B S T R A C TUsing HST WFPC2 data, we estimate half-light radii, morphological classifications and rest-frame, blue-band absolute magnitudes for 270 galaxies with spectroscopic redshifts from z#0 to z:3.43, and 77 thought from their colours to be at 2.0OzO4.5. The mean blue-band surface brightness decreases by #2 mag along the Hubble sequence from ellipticals to irregulars, but peculiar galaxies (forming #21 per cent of the sample) tend to have blue colours but a high surface brightness.The mean rest-frame, blue-band surface brightness increases by 0.95<0.22 mag between z mean %0.2 and z mean %0.9 redshift intervals, with similar evolution for all the morphological types. We estimate that galaxies at 2z3.5 show 2.79<0.31 mag of surface brightness evolution relative to those at z0.35, which is significantly greater than the luminosity evolution over this redshift range. This can be explained by a size and luminosity evolution (SLE) model, in which the outer regions of spiral galaxies form later and with a longer time-scale than the centres, causing the halflight radius to increase with time.The blue-band luminosity density increases by a factor of #2.5 from z#0 to z#2.85, but falls back to approximately its zero-redshift value at z#4. Galaxies at z#4 appear to be similar in surface brightness to those at z#3, but smaller in size and lower in luminosity.The angular size distributions of 22I26 galaxies in the Hubble Deep Field, divided at z#1.5 using UBI colours, suggest that most size evolution occurs at z1.5. A small sample of galaxies with I2K3.8 (passively evolving galaxies at z1) appeared to be no smaller than the PLE prediction, suggesting that it is primarily star-forming galaxies that evolve in size.Galaxy brightness and size evolution to z#4 159 © 1998 RAS, MNRAS 293, 157-176 Figure 1. Distribution of observed I-band surface brightness I for the z2 galaxies in our spectroscopically observed sample (solid histogram) and for galaxies detected in the Groth strip over the same magnitude range (dotted histogram).Galaxy brightness and size evolution to z#4 169 © 1998 RAS, MNRAS 293, 157-176 Figure 10. The observed (U2B)22(B2I) colour index of HDF galaxies against spectroscopic redshift (or colour-estimated redshift for the S96 and M96 galaxies), compared with the PLE models (solid lines). At z:1 the E/S0 (cold) model is at the bottom and the starburst model at the top. The dashed line shows the chosen divide at (U2B)22(B2I):22.5. 172 N. Roche et al. © 1998 RAS, MNRAS 293, 157-176 Figure 13. The distribution of half-light radius r hl , in log arcsec, for HDF galaxies observed in the K band by Hogg et al. (1997), to a limit of K:23 and divided by colour at I2K:3.8, compared with the PLE (dashed line), non-evolving (dotted line), and SLE (dot-dashed line) models.Galaxy brightness and size evolution to z#4 175 © 1998 RAS, MNRAS 293, 157-176