  We measure and analyse the redshift distribution N(z) of magnitude-selected
samples using spectroscopic redshift measurement from the magnitude-selected
VIMOS VLT Deep Survey (VVDS) with 17<iAB<24.75. We compute the N(z) and provide
reference parametric fits for i band, J, H and Ks band magnitud limited
samples. The N(z) of a sample with iAB<24 has a mean redshift z=0.92, with 8.2%
of the galaxies with z>2. Down to iAB<24.75 the sample has a mean redshift
z=1.15 and 17.1% of the galaxies are beyond z=2. The projected sky density is
2.07+/-0.12 gal/arcmin2 at 1.4<z<2.5 and KsAB<22.5, 1.72+/-0.15 gal/arcmin2 at
2.7<z<3.4 and 0.59+/-0.09 gal/arcmin2 at 3.4<z<4.5 brighter than iAB=24.75.
Galaxies at z~3 identified from magnitude-selected samples are 1.5 to 3 times
more numerous than when they are colour-colour selected. We demonstrate that
colour-colour selected samples over 1.4<z<4.5 are strongly contaminated by
galaxies at other redshifts. Semi-analytic models on the Millennium simulations
under-predict the number of luminous star-forming galaxies at z\sim1.8-2, as
well as over-predict the number of low-luminosity galaxies at z<0.8. Our study
provides comprehensive galaxy number counts N(z) from galaxies with
spectroscopic redshifts over a large redshift domain 0<z<5, a solid basis for
the measurement of volume-complete quantities. Magnitude-selected surveys
identify a higher number of galaxies at z>2 than in colour-colour selected
samples, and we use the magnitude-selected VVDS to emphasize the large
uncertainties associated to other surveys using colour or colour-colour
selected samples. Our results further demonstrate that semi-analytical models
on dark matter simulations have yet to find the right balance of physical
processes and time-scales to properly reproduce a fundamental galaxy population
property like the observed N(z).
