  The paper is devoted to a generalized and simplified version of author's
approach to covering theorems in bounded cohomology theory. The amenability
assumptions are replaced by weaker and more natural acyclicity assumprions. In
the case of open coverings the paracompactness assumption is removed. It is
shown that for paracompact spaces the case of closed coverings can be reduced
to the case of open coverings if spaces and subspaces in question behave nicely
with respect to fundamental groups and covering spaces. Another covering
theorem for closed coverings assumes nice behavior with respect to singular
homology; now only its proof uses the sheaf theory. The methods apply also to
$l_1$-homology. The exposition is largely self-contained. In particular, the
required results from the theory of paracompact spaces are presented with full
proofs.
