The category of quasi-coherent sheaves over a scheme, despite being Grothendieck, usually does not have enough projective objects. A possible (and the usual) remedy for this is to consider flat sheaves; this class, however, turns out to be too large for some purposes. The aim of this talk is to present the class of very flat sheaves (or modules in the affine case) as a "minimal" class suitable for doing algebraic geometry and homologically well-behaved at the same time. It turns out that this class has remarkable properties even in the affine setting, e.g. every flat finitely presented commutative \(R\)-algebra is a very flat \(R\)-module. This is joint work with Sergio Estrada and Leonid Positselski.