Morris-Thorne wormholes with a cosmological constant

First, the ideas introduced in the wormhole research field since the work of Morris and Thorne are reviewed, namely, the issues of energy conditions, wormhole construction, stability, time machines and astrophysical signatures. Then, spherically symmetric and static traversable Morris-Thorne wormholes in the presence of a generic cosmological constant $\ensuremath{\Lambda}$ are analyzed. A matching of an interior solution to the unique exterior vacuum solution is done using directly the Einstein equations. The structure as well as several physical properties and characteristics of traversable wormholes due to the effects of the cosmological term are studied. Interesting equations appear in the process of matching. For instance, one finds that for asymptotically flat and anti--de Sitter spacetimes the surface tangential pressure $\mathcal{P}$ of the thin shell, at the boundary of the interior and exterior solutions, is always strictly positive, whereas for de Sitter spacetime it can take either sign as one would expect, being negative (tension) for relatively high $\ensuremath{\Lambda}$ and high wormhole radius, positive for relatively high mass and small wormhole radius, and zero in between. Finally, some specific solutions with $\ensuremath{\Lambda},$ based on the Morris-Thorne solutions, are provided.