  In this paper we consider the following question. What is the maximum number
of pairwise disjoint $k$-spreads which exist in PG(n,q)? We prove that if k+1
divides n+1 and n>k then there exist at least two disjoint k-spreads in PG(n,q)
and there exist at least $2^{k+1}-1$ pairwise disjoint $k$-spreads in PG(n,2).
We also extend the known results on parallelism in a projective geometry from
which the points of a given subspace were removed.
