Grey goo
Grey goo is a hypothesised doomsday scenario in which microscopic, self-replicating robots eat the entire world, leaving only the eponymous grunge coating the entire world (our planet, that is). The idea has been a theme of science fiction, and one or two people are taking it seriously — but even Eric Drexler, populariser of the concept, now thinks it's wrong,[1] and that grey goo scaremongering distracts attention from genuine problems with nanotechnology.
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The problem with the idea
Grey goo is predicated on the sort of thinking common in bad science fiction. In science fiction, the nanotech self replicating robots which eat everything in their path are created in one step. Typically they are miniaturised robots using electronics, which would bring in obvious issues with how one nanobot could construct, program and power up another without exhausting its own energy supply in the process.
This is very distinct from any realistic depiction of technological progress where the first such self-replicators — an enormous breakthrough on their own — would have to sit in a batch of special nutrients and be microwaved or otherwise provided energy to do their work, while being kept perfectly sterile (to keep bacteria from eating your nanotech). Then this technology would gradually be improved in a great many steps and find many uses ranging from cancer cure to dishwashers, with corresponding development in goo control methods.
The levels of metabolic efficiency and sheer universality required for the grey goo to be able to eat everything in its path — and that's stuff which life hasn't already eaten — would require a multitude of scientific and engineering breakthroughs, on top of an incredibly advanced and powerful nanotechnology and nano-manufacturing capacity within artificial environments that include an external energy input and a supply of rare elements for their use as catalysts.
How does an advanced civilization fight the grey goo? We can't know, obviously; but, for example, a goo equivalent of a bacteriophage is going to be way less complicated than the goo itself (as the goo has to be able to metabolize a variety of foods efficiently, while a goophage would only need to metabolize goo, and could even be provided with extra power). Much less dramatically, the delicate chemistry involved in refinement and processing of the raw materials can be disrupted with chemicals that bind to the key catalysts, over-dope the semiconductors, and otherwise interfere with replication - the goo can be poisoned, in other words, with otherwise harmless chemicals in very minute amounts. The speed of goo replication as well as its ability to digest is going to be severely limited by available energy, except for the above-mentioned externally powered nanotech that is disabled with a circuit breaker.
Non-organic materials generally contain little chemical energy (e.g., rocks), so some mechanical goo is unlikely to destroy the world. A goo that could cover the globe relying on the biosphere for energy is far beyond current technology, but it is a possibility that a genetically modified organism could act like an invasive species and upset local environments. This is a little different from "eating everything on the planet," though.
Historical examples
Shockingly, the grey goo scenario has in fact happened in the past. A self-starting and self-replicating nanobot plague covered the Earth! This event is commonly referred to as "life." A later disaster befell most of the life when blue-green algae filled the air with oxygen; this is known as the Great Oxygenation Event
Therefore, you should give funding to chemists who call themselves nanotechnologists! So that they, er, don't do this.
Potential real world examples
There are efforts to make moving vehicles that consist of only a few small particles. In the 1990s, someone theorized that swarms of tiny robots could be used to assemble and disassemble various objects such as buildings, car seatbelts, or even larger robots. There's just a few differences:
1. The utility fog cannot replicate. It has to be mass produced and then released in swarms.
2. The utility fog cannot eat everything. You can cover something in utility fog but it will not produce more utility fog.
3. Most scientists suggest that utility fog should NOT be used to cover the atmosphere. (Seriously, there are debates over this)
4. Given current AI is still unlikely to be able to think for itself, utility fog would be entirely under our control and will NOT attempt to consume us all.