A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ2PG(2,q))≤ 2(q+(q-1)/(r-1)). For a finite projective plane Π, let denote the maximum number of classes in a partition of the point-set, such that each line has at least two points in some partition class. It can easily be seen that (⋆) for every plane Π on v points. Let , p prime. We prove that for , equality holds in (⋆) if q and p are large enough.