A fano variety is a smooth, geometrically connected variety over a field, for which the dualizing sheaf is anti-ample. For example the projective space, more generally flag varieties are Fano varieties, as well as hypersurfaces of degree d ≤ 𝑛 in ℙ𝑛. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those invariants. We present a geometric version of it.