Disadvantages
On the downside, staging requires the vehicle to lift motors which are not being used until later, as well as making the entire rocket more complex and harder to build. In addition, each staging event is a significant point of failure during a launch, with the possibility of separation failure, ignition failure, and stage collision. Nevertheless the savings are so great that every rocket ever used to deliver a payload into orbit had staging of some sort.
In more recent times the usefulness of the technique has come into question due to developments in technology. In the case of the Space Shuttle the costs of space launches appear to be mostly composed of the operational costs of the people involved, as opposed to fuel or equipment. Reducing these costs appears to be the best way to lower the overall launch costs. New technology that is mainly in the theoretical and developmental stages is being looked at to lower the costs of launch vehicles. More information can be found on single stage to orbit designs that do not have separate stages.
Read more about this topic: Multistage Rocket