The following is something that I find myself clarifying for people frequently so I thought it was worthwhile producing it here.
Greyhound Rescue is a number of individual charities and good-causes that rescue and re-home greyhounds and lurchers. There is no official body regulating them, and all of their income has to be earned by charitably fund raising. These groups vary in size between large, well structured groups that cover most of the country, to a couple of people operating around full time jobs. These groups all operate independently although their geographical areas may overlap. Most groups have good informal relationships, for instance passing on a prospective home if they don't have a suitable dog at that time.
In addition to ex-racing dogs, most Greyhound Rescue groups will accept for rehoming greyhounds with no ear-marks, i.e. never registered for the race track, and also lurchers. A 'working' definition of a lurcher is a dog with a visible component of greyhound or other sight hound parentage.
The individual Greyhound Rescue groups decide their own operating 'rules'. For instance, some groups may rehome a suitable dog to people working full time, while another group might never rehome any dog to such a home. Similarly there will be different views on rehoming dogs to families living in flats, or any number of other issues.
www.greyhoundrescue.co.uk is not a charity, a rescue/rehoming group or a regulatory body. The original idea for the website was to provide a centralised listing of as many Greyhound Rescue groups as possible, at a time when few had their own websites. Now the main function of this site is to link to UK Greyhound Rescue charity and good-cause websites, and also to nurture and support interest in greyhound issues.
The Retired Greyhound Trust (RGT) is different from the Greyhound Rescue groups described above. The RGT is partly sponsored by the regulated racing industry (as opposed to the non-regulated 'flapping' tracks) and partly by charitable fund raising. Most regulated race tracks have an RGT branch that works to re-home as many dogs as possible after their retirement from racing. The majority of these groups will only accept for homing dogs from the track.
Greyhound Rescue groups and the RGT branches are all heavily over subscribed with dogs needed rehoming. All groups are in the position that they sometimes have to turn dogs away because they do not have enough money/kennel space/foster homes. For these dogs it is often the 'end of the line'.