  We study the question how many subgroups, cosets or subspaces are needed to
cover a finite Abelian group or a vector space if we have some natural
restrictions on the structure of the covering system. For example we determine,
how many cosets we need, if we want to cover all but one element of an Abelian
group. This result is a group theoretical extension of the theorem of Brouwer,
Jamison and Schrijver about the blocking number of an affine space. We show
that these covering problems are closely related to combinatorial problems,
including the so called additive basis conjecture, the three-flow conjecture,
and a conjecture of Alon, Jaeger and Tarsi about nowhere zero vectors.
