Older rats are prone to tumours; in particular about 1 in 3 does (and a small proportion of bucks) will develop mammary tumours. Most mammary tumours in British rats seem to be lipomas, arising from the fat in the mammary gland rather than the gland itself, and are encapsulated, benign and immediately behind a nipple: but there are also some more dangerous tumours which occur in the glandular tissue itself and these can occur in outlying fronds of gland, quite far from the associated nipple. American strains are said to be prone to endocrine tumours arising in the mammary tissue itself, and these are more likely to be aggressive than are lipomas.
The doe has 12 nipples, but really only 8 breasts. In front of her fore-legs and just behind/between her hind-legs are pairs of breasts, each with a single nipple. Behind her fore-legs and in front of her hind-legs are pairs of larger breasts with two nipples: I don't know if there's a separate mammary gland for each nipple and they are just very close together, or whether these double-nippled breasts only contain one mammary gland. For purposes of "going lumpy" they behave as if they have only one gland, so a doe can develop a maximum of eight mammary tumours.
Metastasis does not seem to occur in Norway rats (and is rare in mice) provided the immune system is sound. A few types of tumour are invasive - that is, they spread in situ and infiltrate surrounding tissue - but you don't get secondary tumours cropping up at sites physically remote from the primary one. It's a moot point whether any tumours in rats can truly be classed as cancerous. Malignant tumours - cancers - are normally defined as those which are invasive and metastasizing, while benign tumours are those which are neither invasive nor metastasizing. There isn't really a word for tumours which are invasive but not metastasizing, as it's a situation which wouldn't normally occur in humans.
Inoperable internal tumours of e.g. the brain and lungs do occasionally occur. But the great majority of tumours in rats are benign, painless and outside on their skin or just under it. Most - including the great majority of mammary tumours - can easily be removed if they start getting in the animal's way. Rats survive both anaesthetic and operations very well, with little distress; so the majority of tumours present little more medical difficulty than an abscess. However when choosing your rats you may like to remember that does are likely to represent a higher veterinary cost.
Because metastasis almost never occurs in Norway rats there is no great hurry about removing tumours, and it is generally safe to leave them until they start getting in the animal's way. Watch out, however, for an unpleasant soft, squishy tumour occurring alongside the vagina and anus in does. This type of tumour grows rapidly and sometimes becomes wrapped around the urethra, making it inoperable. Even very large examples of this tumour can usually be removed safely by a skilled vet, but the risk of it becoming inoperable is much higher than for mammary tumours: so it should be removed a.s.a.p. unless the rat is so old and frail that surgery is a major risk.
This tumour can also mislead vets into thinking it is inoperable even when it isn't. It often feels, and moves, as if it is attached to the rectum, and your vet may therefore decide the situation is hopeless: but in fact if the tumour is operated on it always or nearly always peels away from the rectum without difficulty. Indeed, since metastasis hardly ever occurs in rats, it is very unlikely that this tumour (which is fatty) could spread to the rectum (which is epithelial tissue). I suspect what it is doing is infiltrating the fat in the mesenteric membrane which links sections of gut, so it moves with the rectum because it's stuck to the membrane which is attached to the rectum, but it can quite easily be cut out.
There are other kinds of aggressive tumour which are never or almost never operable, such as tumours of the nasal bone or ear canal, papilloma, various skin tumours and one arising as a progressive thickening of the skin around the mammary glands. Tamoxifen may reduce those of glandular origin, but otherwise you will just have to treat the symptoms for as long as the animal is taking an interest in life and not seriously distressed, and then put it down.
Internal tumours are rare, at least in British rats. For symptoms of brain tumours see under Neurological Disorders.
It is extremely difficult to eliminate tumours by selective breeding and still end up with a healthy rat. More often than not, it's the fit, glossy, vigorous rat who breaks out in lumps, and the scrawny, wheezy one with incipient kidney-failure who remains tumour-free. In part this may be because many rat tumours arise from somatic mutations of fatty tissue, and unhealthy rats have less fatty tissue to mutate: but I suspect that growing tumours is of itself a sign of vigour. This would make sense. Rats retain the ability to grow new tissue after injury (including new nerve tissue following strokes), and to grow in size throughout their lives, to a much greater extent than most other mammals do: but having an increased ability to grow new tissue probably entails an increased risk of growing new tissue you don't want, i.e. tumours. Scrawny, sickly rats usually have a reduced ability to heal and to grow - and that probably means a reduced risk of growing extra bits.
For anyone who is trying to breed healthy rats, I would say don't breed from any strain which has a high incidence of non-mammary tumours/tumours occurring in bucks, or which has an excessively high incidence of mammary tumours (if your does are averaging more than two each before two years old, you probably have a problem). But if it gets down to a choice between a bouncy, glossy strain with a few lumps and a sad skinny one which is lump-free, put up with the lumps: they are only a strain on your pocket, whereas kidney-failure is a strain on the rat.