Context.The VIMOS VLT Deep Survey (VVDS) was undertaken to map the evolution of galaxies, large scale structures, and active galaxy nuclei from the redshift spectroscopic measurements of ~105 objects down to an apparent magnitude $I_{AB} = 24$, in combination with a multi-wavelength acquisition for radio, infrared, optical, ultraviolet, and X-rays data. Aims.We present the evolution of the comoving star formation rate (SFR) density in the redshift range $0 < z < 5$ using the first epoch data release of the VVDS, that is 11564 spectra over 2200 arcmin2 in two fields of view, the VVDS-0226-04 and the VVDS-CDFS-0332-27, and the cosmological parameters ($\Omega_\mathrm{M}$, $\Omega_{\Lambda}$, $h)=(0.3$, 0.7, 0.7). Methods.We study the multi-wavelength non dust-corrected luminosity densities at $0 < z < 2$ from the rest-frame far ultraviolet to the optical passbands, and the rest-frame 1500 Å luminosity functions and densities at $2.7 < z < 5$. Results.They evolve from $z=1.2$ to $z=0.05$ according to $(1+z)^{x}$ with $x = 2.05, 1.94, 1.92, 1.14, 0.73, 0.42$, and 0.30 in the FUV-1500, NUV-2800, U-3600, B-4400, V-5500, R-6500, and I-7900 passbands, respectively. From $z=1.2$ to $z=0.2$ the B-band density for the irregular-like galaxies decreases markedly by a factor 3.5 while it increases by a factor 1.7 for the elliptical-like galaxies. We identify several SFR periods; from $z=5$ to 3.4 the FUV-band density increases by at most 0.5 dex, from $z=3.4$ to 1.2 it decreases by 0.08 dex, from $z=1.2$ to $z=0.05$ it declines steadily by 0.6 dex. For the most luminous $M_{AB}(1500~\AA)<-21$ galaxies the FUV-band density drops by 2 dex from $z=3.9$ to $z=1.2$, and for the intermediate $-21<M_{AB}(1500~\AA)<-20$ galaxies it drops by 2 dex from $z=0.2$ to $z=0$. Comparing with dust corrected surveys, at $0.4 \la z \la 2$ the FUV seems obscured by a constant factor of ${\sim}1.8$–2 mag, while at $z<0.5$ it seems progressively less obscured by up to ${\sim}0.9$–1 mag when the dust-deficient early-type population is increasingly dominating the B-band density. Conclusions.The VVDS results agree with a downsizing picture where the most luminous sources cease to efficiently produce new stars 12 Gyrs ago (at $z\simeq 4$), while intermediate luminosity sources keep producing stars until 2.5 Gyrs ago (at $z\simeq 0.2$). A modest contribution of dry mergers and morphologies evolving towards early-type galaxies might contribute to increase the number density of the bright early types at $z<1.5$. Our observed SFR density is not in agreement with a continuous smooth decrease since $z\sim4$.