The present monograph concerns with the invariant subspace problem within the context of families of linear bounded operators on Banach spaces as well as families of positive operators on Banach lattices. Although it is primarily based on the PhD thesis of the author, the monograph provides further results and information on recent advances about the topic. The general invariant subspace problem concerns linear and bounded operators on infinite-dimensional complex Hilbert spaces, and asks whether there exists a closed subspace, different than the trivial ones, that is mapped to itself by such an operator. The picture that emerged after oodles of different attempts for a possible solution to this problem for a single operator has been the subject matter of an extensive amount of research over more than seven decades, and it was only in the late 1990s that the reformulation of the invariant subspace problem for families of operators, rather than a single one, attracted the mathematical community''s growing attention. In this book, we are interested in giving some results concerning the invariant subspace problem for special families of operators.