A linear operator T: X → Y between vector lattices is said to be disjointness preserving if T sends disjoint elements in X to disjoint elements in Y. Two closely related questions are discussed in this paper: (1) If T is invertible, under what assumptions does the inverse operator also preserve disjointness? (2) Under what assumptions is the operator T regular? These problems were considered by the authors in  but the current paper (closely related to  but self-contained) reflects the progress made during the last few years. In particular we completely describe the vector lattices on which every disjointness preserving operator is regular, obtain very general conditions which guarantee that the inverse operator preserves disjointness, and introduce a very large class of vector lattices which we call weakly c₀-complete (and which contains in particular all relatively uniformly complete vector lattices); for the lattices from this class we obtain almost complete answers to the above-mentioned questions. Additional topics include a considerable improvement of Huijsmans-de Pagter-Koldunov's theorem and applications to the spaces of continuous functions.