Just returned from the excellent 2nd EmergeNet meeting, in Glasgow. EmergeNet is an EPSRC funded network of projects and people linked by an interest in emergence. As the Wikipedia article states, the phenomenon of emergence has been known about for a long time, but it still defies a proper scientific definition. In other words a definition that allows you to look at some complex phenomena and say yes, this is true emergence, but that isn't, and to measure the strength of the emergence (if indeed that is possible).
The reason a rigourous definition of emergence is important is that we can now contemplate designing complex systems that exploit emergence. A swarm robotics system is, for instance, a designed system which relies on emergence but - within the framework of complexity science - many other systems, from molecular to economic, would benefit from a deep understanding of emergence.
There were some truly excellent talks at EmergeNet2 - I'll add a link here when the presentations are online. But from one of those talks here is a link to an astonishing YouTube video from EmergeNet leader Lee Cronin and his team, showing (if I understand it correctly) controlled inorganic crystalline growth of molecular tubes - which looks remarkably organic.