Since this is a part of the defining sustainability series, I am adding my own perspective here. Other perspectives of course are welcome and the whole purpose is to provide public insight into what's on and off topic here so please vote on these.
The problems @Pitt noted above are design issues with well understood solutions, which include biodiversity for starters (so that removing part of one species enables fallback on others). The larger barrier of course is knowledge. Permaculture involves working smarter rather than harder, and so reducing the overall energy inputs. This means humans have to do most stuff, you can't leave it to machines.
I think that permaculture is an important aspect of any attempt to become sustainable because it is sustainable itself, because we must mimic natural ecosystems in our own approaches to material and waste management, and because it cultivates interconnected and sustainable communities on all levels. This is not to say that there aren't challenges regarding human inputs and knowledge levels, and that these may have impacts elsewhere. On the whole, I think that widespread permaculture is a good thing however.
Permaculture is Sustainable Itself
Permaculture is sustainable with few external inputs. Permaculture utilizes processes of evolutionary ecology to replace most of the external direct energy inputs aside from the sun, as well as all indirect inputs beyond the initial design and implementation phase. This largely cuts fossil fuels out of the system of food production.
Permaculture utilizes carefully chosen plants and domestic animals, as well as sometimes wildlife, for pest control. For example one might plant thorny hedges to keep deer out of the food production area, while providing habitat for birds of prey so that that they can prevent rodents from becoming a problem. Pests an never be entirely eliminated because predators always lag behind pests, but in this way they can be controlled so that they do not cause substantial damage in established permacultures. A good beginner's book on this subject is Gaia's Garden.
A second point is that species are arranged in ways which mutually support eachother. I haven't been able to experiment with this but a example of an area where this might be helpful might be co-raising cattle and horses, together in an area that has snow so as to minimize the need for hay. Cattle won't eat what they can't see so they tend to starve quickly when it snows, but horses will kick the snow away to eat. Thus the horses can be used to help the cattle forage in winter. If areas are maintained with sufficient forage, hay may not even be necessary (there are recorded cases of cowboys with herds stuck in winter storms following mustang herds in order to ensure the cattle could eat, and thus saving their herds).
The Necessity for Material Lifecycles to Mimic Ecosystems
In an ecosystem, the waste product of one species or element becomes the input of another. Bears poop in the woods. Micro-organisms break down the poop and they release waste products which plants take in. Those plants may produce fruit and the bears eat the fruit. The bears then poop and the cycle repeats. This example shows the way in which nutrient cycles work in ecosystems, re-using and re-cycling all components and over time building quality soils and very efficient ways of transforming current solar energy into biological energy.
Sustainability requires, among other things, that we stop polluting our environment with things which cannot readily be re-used in the biological context. For this reason it is very important to develop closed material lifecycles. Our current factory-to-landfill must eventually be replaced with a closed cycle where very little if anything ends up in land fills because everything is re-used in some context or another. Doing this requires in essence having the output or waste of one system becoming the input of another. Recycling is one way this can happen (re-use is another), but either way waste must be re-purposed as an asset. An example might be privatizing sewage treatment, not as a waste disposal service, but as a series of cooking gas and fertilizer factories. Waste products which cannot be recycled can be re-used in unexpected ways, such as building things out of "urbanite."
The shift in thinking from a factory to landfill pipeline to a series of interconnected and infinitely continuous series of material lifecycles (akin to the water cycle, or to nutrient cycles in ecosystems), is absolutely necessary to achieving sustainability, and permaculture teaches this quite a bit. This is a reason why I am such of a fan of urban permaculture. While the idea of feeding a family of 4 on a city lot is probably unrealistic, the lessons learned about sustainability and the increase in free time from doing so (instead of high maintenance flower gardens!) is well worth it. Most of my "flower gardens" also produce fruits and veggies, they look nice, and they require very, very little maintenance.
Sustainable Communities
The emphasis on the output of one component being the input of another has the potential at least to build close-knit, sustainable communities through garden exchanges, and also things like thinking about what one can re-use things that would otherwise be discarded. Urbanite as a building material for small projects (say outdoor pizza ovens) comes to mind but the possibilities here are really endless. The more we think about what we can do with waste, the less of it ends up being discarded in unsustainable ways and the closer we become as communities as we utilize the waste products of neighbors and vice versa,
Sustainability Challenges
The big one actually is manpower (and this is an issue for most green energy too btw but for very different reasons), and the second is education. Building permacultures requires years of practice and/or significant training. It also turns food production into an intellectually challenging activity. However it is also less physically challenging than conventional agriculture in many cases, and so there are possibilities of closing loops here. The problem with manpower though is not so easily overlooked. With modern fossil-fuel-based agriculture it is not hard for an individual to successfully farm vast tracts of land, possibly in the hundreds of acres, with only seasonal help. Permaculture requires more people to be working in food production. It also requires that people make more careful choices regarding foods to consume (grains are less sustainable than chestnuts for example, and wheat is less sustainable than rice). More people in "farming" full-time means fewer people engineering green energy solutions and the like. Competition for manpower is thus a significant concern as I see it.
TL;DR
I see permaculture as pretty close to a necessity for achieving sustainability both due to its direct effects and also its indirect effects. I also see permaculture ultimately as the best current training ground for the sorts of patterns we must develop in order to achieve sustainability generally.