Grey goo refers to a hypothetical 
end-of-the-world event involving 
molecular nanotechnology in which out-of-control 
self-replicating robots consume all living matter on 
Earth while building more of themselves (a scenario known as 
ecophagy).
  The term is usually used in a 
science fiction context. In a worst-case scenario, all of the matter in the 
universe could be turned into goo (with "goo" meaning a large mass of replicating nanomachines lacking large-scale structure, which may or may not actually appear 
goo-like), killing the universe's residents. The disaster is posited to result from an accidental 
mutation in a self-replicating nanomachine used for other purposes, or possibly from a deliberate 
doomsday device.
  
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Definition of grey goo
  The term was first used by molecular nanotechnology pioneer 
Eric Drexler in his book 
Engines of Creation (1986). In Chapter 4, 
Engines Of Abundance, Drexler explores a scary scenario of 
exponential growth with such 
assemblers:
   "Thus the first replicator assembles a copy in one thousand seconds, the two replicators then build two more in the next thousand seconds, the four build another four, and the eight build another eight. At the end of ten hours, there are not thirty-six new replicators, but over 68 billion. In less than a day, they would weigh a ton; in less than two days, they would outweigh the Earth; in another four hours, they would exceed the mass of the Sun and all the planets combined - if the bottle of chemicals hadn't run dry long before."   Drexler describes grey goo in Chapter 11 
Engines Of Destruction:
   "...early assembler-based replicators could beat the most advanced modern organisms. "Plants" with "leaves" no more efficient than today's solar cells could out-compete real plants, crowding the biosphere with an inedible foliage. Tough, omnivorous "bacteria" could out-compete real bacteria: they could spread like blowing pollen, replicate swiftly, and reduce the biosphere to dust in a matter of days. Dangerous replicators could easily be too tough, small, and rapidly spreading to stop - at least if we made no preparation. We have trouble enough controlling viruses and fruit flies."   It is thus worth noting that grey goo need not be grey or gooey. They could be like, for all purposes, a 
plant or 
bacteria. It is only the result of their ecophagy that would resemble grey goo.
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Green goo
  One convenient analogy for the grey goo problem is to consider bacteria as the most perfect example of biological nanotechnology. As they have not reduced the world to 
green goo, some consider it unlikely that some artificial construct will manage to do so with grey goo.
  Even so, some people argue that green goo, or even a combination of 
nanotechnology and 
biotechnology to create organic 
replicators, is a more realistic threat than grey goo. Arguing that bacteria are 
ubiquitous and extraordinarily powerful, 
Bill Bryson (2003) says that the Earth is "their planet" and that we're only allowed to exist on it because "they allow us to". Margulis and Sagan (1995) go further, arguing that all organisms, having descended from bacteria, 
are in a sense bacteria. Many kinds of bacteria are in fact essential for human life and are found in large quantities in the human digestive tract, in a 
symbiotic relationship.
  Thus a green goo could be a 
multicellular organism that obtains its raw materials to grow through ecophagy, and then grows through a process of 
exponential assembly such as 
cell division