Most no-kill shelters are limited access facilities, which means that they only accept animals when they have resources available for their care. Some of the shelters that I am affiliated with are no-kill. They are networked with several other organizations, so animals that stay at the shelter for several weeks without any adoption interest are transported to a different shelter within the network, where their chances for adoption are greatly improved. This system has worked very well. The key to actually helping these animals is to get them placed quickly because being incarcerated for a long period of time in a shelter enviornment is not good for them mentally or physically. Transporting has been a blessing because animals that are not very adoptable in one part of the State may be highly desirable in another. One of the local kill shelters has also joined the transport network and they have reduced the number of animals that they euthanize by about 50%. Animals that are still not adopted after making the rounds usually wind up in a foster home or at a sanctuary like mine. The bad thing about a no-kill is that they have to turn animals away if they do not have room for them, but the good thing is that the animals that do find themselves in one are going to wind up in a good home, the key is to get it done in a timely fashion.
Edited to add that I agree with Summitview...the no-kill terminology is often a misnomer...I don't know of any shelter that will not have an animal put to sleep if they are very sick or very agressive...most screen so they rarely get a mean or sick animal, but the term low-kill is actually more accurate.