Let PG(r, q) be the r-dimensional projective space over the finite field GF(q). A set Χ of points of PG(r, q) is a cutting blocking set if for each hyperplane Π of PG(r, q) the set Π ∩ Χ spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes. Of particular interest are those having a size as small as possible. In this talk, I will discuss known constructions of cutting blocking sets, from which there arise minimal linear codes whose length grows linearly with respect to their dimension. I will also present two new constructions of cutting blocking sets whose sizes are smaller than the known ones.