One of the more exciting possibilities is barely electronic at all. Roxatane devices are more mechanical than anything else: a combination of a circular molecule wrapped around another that acts as a spindle, with end stops that prevent the whole thing drifting apart. Unlike normal chemical compounds, the structure is kept together by purely mechanical means rather than the usual shared electron bonds, so it can change state rapidly without the sort of energies normally involved in breaking apart and creating compounds.
Roxatanes -- and their close relatives, the linked ring catenates -- have been known about since the early 1980s, but they were produced in very small quantities by chance during other reactions and seen as exotic curiosities of no real use. Advances in mechanical chemistry have changed that: they can now be mass-produced cheaply and reliably by reacting chemicals together on templates that hold the components in the right alignment to create the finished article.
With the right structures, roxatane molecules can be built to be bistable -- like a light switch, they can remain in one of two stable states. You only need energy to switch between them. If one of the states has a higher electrical resistance than the other, the molecule becomes a memory device that can store a one or a zero that can be read by electronic circuits. By arranging the molecule so that it can flip between the states when it gets the right electrical pulse, a complete device can be built that's only a few hundred atoms big.
This is where HP Labs and others get excited. Create two sets of parallel wires -- one running north-south, the other east-west -- and put roxatane molecules at each cross-over, and you have a memory architecture that can theoretically shrink much further than anything depending on transistors -- achieving a feature size of two to three nanometres. It should also be less power hungry and faster. The semiconductor physics underpinning transistor design has limits on how small voltages can be before they stop being effective: different and more advantageous rules apply for molecular switches.