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The galaxy Luminosity Function (LF) has been estimated in the rest frame Bluminosity at 0<z<1.25 and at 1700 {\AA} for 2.5<z<4.5 from deep multicolorsurveys in the HDF-N, HDF-S, NTT-DF. The results have been compared with arecent version of galaxy formation models in the framework of hierarchicalclustering in a flat Cold Dark Matter Universe with cosmological constant. Theresults show a general agreement for z<= 1, although the model LF has a steeperaverage slope at the faint end; at z~3 such feature results in anoverprediction of the number of faint (I_{AB}~ 27) galaxies, while theagreement at the bright end becomes critically sensitive to the details of dustabsorption at such redshifts. The discrepancies at the faint end show that arefined treatement of the physical processes involving smaller galaxies is tobe pursued in the models, in terms of aggregation processes and/or stellarfeedback heavily affecting the luminosity of the low luminosity objects. Theimplications of our results on the evolution of the cosmological star formationrate are discussed.Comment: Revised version; corrected magnitudes at 1700 Angstrom in figure 2;  ApJ

The Evolution of the Luminosity Function in Deep Fields: A Comparison with Cold Dark Matter Models