Even though Norway rats are strong swimmers most rats dislike being bathed. They usually learn to tolerate it once they realize you aren’t trying to drown them; but it still amounts to “force-grooming” which they are likely to interpret as aggressive behaviour, so it’s not a thing which should be done unnecessarily.
Rats should be bathed under the following sort of circumstances:
a): Dipping them in anti-fungal or anti-mite preparations to clear skin conditions
b): Old bucks sometimes have problems with their scent-glands and develop a thick orange grease all through the fur on their back, which is uncomfortable and sticky and needs to be cleaned off with a strong de-greasing shampoo (this is more complicated than normal bathing: see next section for details of how to do it)
c): Some old or sick rats can't chew solid food and have to be fed things like baby food and cooked oatmeal - if they are very decrepit they may get gunged up with this and not be able to clean it off themselves
d): Rats who are more or less paralyzed at the back end will often end up urinating over their own thighs and stomach, so the affected areas will need to be rinsed every day and treated with a non-greasy nappy-rash cream
e): Some groups of rats have an unlovely habit of peeing on each other in the nest, and become so sticky and smelly as a result that they are actively unpleasant to handle
f): Some rats do daft things like frolicking in the fireplace and coming out covered in soot, and it is probably better to rinse it off in the sink rather than to let them lick it off
g): If you want to show your rats and they are not of the cleanest (pale rats in particular often develop orange stains between the shoulders where their cagemates have drooled on them during grooming sessions - though this can sometimes be cleaned up just by dabbing the mark with some of the tear-stain remover which petshops sell for use on runny-eyed white dogs)
The easiest way to do it is usually with running water/a shower-spray, warm but not hot: moisten the rat, rub in the shampoo and then rinse off by standing them under the tap/spray, rather than by trying to dunk them in deep water. However, I used to have an old mink variegated doe called Posy who had been left very disabled after a stroke, as a result of which she was on a cooked-oatmeal diet which she tended to roll through which meant she had to be bathed every day, and she liked to lie on her back in a sink full of warm water and float, with just my finger under the back of her neck to keep her nose above water.
Dirty tails may be cleaned with a little shampoo and a soft toothbrush or nailbrush. Don't scrub hard from the end of the tail towards the body, as this is likely to rub the scales the wrong way and be uncomfortable.
Elderly bucks often develop a thick layer of orange grease in the fur on their back, due to problems with their scent glands. This is harmless, and can be removed by bathing occasionally with a non-toxic de-greasing cleanser such as Swarfega or a strong shampoo (this makes the coat dry, so follow it with a nice conditioner!). If using Swarfega, try if possible to stand the tub by the fire to take the chill off it first: nobody likes having ice-cold slime slapped on their backs without warning, and this is a really good way to get yourself bitten. The cleanser will need to be massaged into the coat quite vigorously to dislodge scales of wax from the skin: if using Swarfega apply it initially to a dry coat not a wet one, but be sure to rinse it out thoroughly when you've finished. You may need two or three cycles of wash-and-rinse to shift a heavy build-up of grease.
Bucks who suffer from this sort of build-up of grease may also develop small sebaceous cysts. The ones I've seen were all harmless: they either remained as a little blip, or opened up so that the pellet of matter inside could be sqeezed out easily. However I am told that in some case they can become irritating and/or infected - perhaps as a result of a particular climate, or central heating - and if your rat is prone to this problem it may be advizable to empty these cysts as soon as they appear. Annette Rust of Virginia has found that a warm compress softens them and makes them easier to express.
Scabs in the coat can be due to a wide variety of causes, including fungal or mite infections (see section on infection), too much protein in the diet, allergy, or cutting themselves with their own claws and not noticing (most rats don't have very sensitive skins). Trim the points off the hind claws if they seem very long and/or sharp (being careful not to cut right down to the quick - the pink line of vein inside the claw), and then work through the other causes by process of elimination, i.e. if it doesn't respond to treatment for mites or ringworm try reducing the amount of protein in the diet, changing the bedding etc..
Some rats have a behavioural problem called "barbering", in which they chew holes in their own and/or their cagemates' fur. Shaving their own forearms is favourite. This is harmless but it ruins the coat for display purposes: the behaviour is inherited, so if you are thinking of showing your rats you should never breed from a barber.
Elderly rats often develop bald spots on the back - see care of the geriatric rat. Rexes are liable to lose most of the coat off their back in late middle age.
All rats have loose skin (a great boon when closing the incision after operations), but it's only when their fur starts to thin that you can see that, while some rats are just a bit baggy, others are covered in corrugated rolls and folds and look like a pink concertina.
Hairless rats (which are not kept in the UK) can develop sore patches in the folds of their skin and need to be lubricated with a little baby-talc or similar.