We show that the mass-metallicity relation observed in the local universe is due to a more general relation between stellar mass M _ { \star } , gas-phase metallicity and star formation rate ( SFR ) .
Local galaxies define a tight surface in this 3D space , the Fundamental Metallicity Relation ( FMR ) , with a small residual dispersion of \sim 0.05 dex in metallicity , i.e , \sim 12 % .
At low stellar mass , metallicity decreases sharply with increasing SFR , while at high stellar mass , metallicity does not depend on SFR .

High redshift galaxies , up to z \sim 2.5 are found to follow the same FMR defined by local SDSS galaxies , with no indication of evolution .
In this respect , the FMR defines the properties of metal enrichment of galaxies in the last 80 % of cosmic time .
The evolution of the mass-metallicity relation observed up to z=2.5 is due to the fact that galaxies with progressively higher SFRs , and therefore lower metallicities , are selected at increasing redshifts , sampling different parts of the same FMR .

By introducing the new quantity \mu _ { \alpha } = log ( M _ { \star } ) - \alpha log ( SFR ) , with \alpha =0.32 , we define a projection of the FMR that minimizes the metallicity scatter of local galaxies .
The same quantity also cancels out any redshift evolution up to z \sim 2.5 , i.e , all galaxies follow the same relation between \mu _ { 0.32 } and metallicity and have the same range of values of \mu _ { 0.32 } .
At z > 2.5 , evolution of about 0.6 dex off the FMR is observed , with high-redshift galaxies showing lower metallicities .

The existence of the FMR can be explained by the interplay of infall of pristine gas and outflow of enriched material .
The former effect is responsible for the dependence of metallicity with SFR and is the dominant effect at high-redshift , while the latter introduces the dependence on stellar mass and dominates at low redshift .
The combination of these two effects , together with the Schmidt-Kennicutt law , explains the shape of the FMR and the role of \mu _ { 0.32 } .
The small metallicity scatter around the FMR supports the smooth infall scenario of gas accretion in the local universe .