Metallicity appears to be one the most important tool to study formation and evolution of galaxies. Recently, we have shown that metallicity of local galaxies is tightly related not only to stellar mass, but also to star formation rate (SFR). At low stellar mass, metallicity decreases sharply with increasing SFR, while at high stellar mass, metallicity does not depend on SFR. The residual metallicity dispersion across this Fundamental Metallicity Relation (FMR) is very small, about 0.05dex. High redshift galaxies, up to z~2.5, are found to follow the same FMR defined by local SDSS galaxies, with no indication of evolution. At z>2.5, evolution of about 0.6dex off the FMR is observed, with high-redshift galaxies showing lower metallicities. This result can be combined with our recent discover of metallicity gradients in three high redshift galaxies showing disk dynamics. In these galaxies, the regions with higher SFR also show lower metallicities. Both these evidences can be explained by the effect of smooth infall of gas into previously enriched galaxies, with the star-formation activity triggered by the infalling gas. ; Comment: Invited talk at the 54th SAIt congress, Naples, 4-7 May 2010