We present u,b,r & i galaxy number counts and colours both from the North and South Hubble Space Telescope Deep Fields and from the William Herschel Deep Field .
The latter comprises a 7 ^ { \prime } \times 7 ^ { \prime } area of sky reaching b \sim 28.5 at its deepest .
Following Metcalfe et al .
( 1996 ) we show that simple Bruzual & Charlot evolutionary models which assume exponentially increasing star-formation rates with look-back time and q _ { 0 } =0.05 continue to give excellent fits to galaxy counts and colours in the deep imaging data .
With q _ { 0 } =0.5 , an extra population of ‘ disappearing dwarf ’ galaxies is required to fit the optical counts .

We further find that the ( r - i ) : ( b - r ) colour-colour diagrams show distinctive features corresponding to two populations of early- and late-type galaxies which are well fitted by features in the Bruzual & Charlot models .
The ( r - i ) : ( b - r ) data also suggest the existence of an intrinsically faint population of early-types at z \sim 0.1 with similar properties to the ‘ disappearing dwarf ’ population required if q _ { 0 } =0.5 .

The outstanding issue remaining for the early-type models is the dwarf-dominated IMF which we invoke to reduce the numbers of z > 1 galaxies predicted at K < 19 .
For the spiral models , the main issue is that even with the inclusion of internal dust absorption at the A _ { B } = 0.3 mag level , the model predicts too blue ( u - b ) colours for late-type galaxies at z \sim 1 .
Despite these possible problems , we conclude that these simple models with monotonically increasing star-formation rates broadly fit the data to z \sim 3 .

We compare these results for the star formation rate history with those from the different approach of Madau et al .
( 1996 ) .
We conclude that when the effects of internal dust absorption in spirals are taken into account the results from this latter approach are completely consistent with the \tau =9Gyr , exponentially rising star formation rate density out to z \approx 3 which fits the deepest , optical/IR galaxy count and colour data .

When we compare the observed and predicted galaxy counts for UV dropouts in the range 2 \la z \la 3.5 from the data of Steidel et al .
( 1999 ) , Madau et al .
( 1996 ) and new data from the Herschel and HDF-S fields , we find excellent agreement , indicating that the space density of galaxies may not have changed much between z = 0 and z = 3 and identifying the Lyman break galaxies with the bright end of the evolved spiral luminosity function .
Making the same comparison for B dropout galaxies in the range 3.5 \la z \la 4.5 we find that the space density of intrinsically bright galaxies remains the same but the space density of faint galaxies drops by a factor of \sim 5 , consistent with the idea that L ^ { * } galaxies were already in place at z \approx 4 but that dwarf galaxies may have formed later at 3 \la z \la 4 .