Also see section on the wild rat, which includes advice on how to foster an orphaned wild rat without risking the health of your pet rats.
One often sees scare-stories in the press about the danger of catching a potentially fatal infection called Leptospirosis (a.k.a. Weil's Disease) from rats. There has never been a recorded case of Leptospirosis in a domestic rat in Britain - but wild rats are a different matter. If you find yourself e.g. caring for an orphaned wild rat it should have a blood-test for Leptospirosis and be quarantined away from your domestic rats for several weeks, so that any infections and/or parasites it may be carrying can be identified and cured with appropriate medication.
If you yourself develop 'flu'-like symptoms after coming into contact with the urine of any wild rodent you should ask your doctor for antibiotics immediately: Weil's Disease isn't dangerous if treated early enough. This applies also to the urine of cattle (though one never sees scare-stories in the press about nasty dirty cows). Cattle carry a closely related condition called Leptospirhardja, which is actually more infectious than Leptospirosis, though not quite so dangerous once you've got it.
Weil's Disease is contracted when the urine of an infected cow or wild rodent gets into an open lesion - so if you have any raw cuts or grazes on your skin it is as well to cover them with something waterproof before coming into contact with any body of water which looks a bit lived-in.
Wild rats in North America can pass on Hantavirus pulmonary syndrome, a sort of pneumonia which begins with fever and muscle aches 1-5 weeks after exposure, followed by shortness of breath and coughing severe enough to require hospitalization, and often ventilation within 24 hours. This is a serious problem, as Hantavirus is harder to treat than Leptospirosis, and is sometimes fatal. However the main vector for Hantavirus is the deer mouse: it's very rare even in wild rats (it's not common even in the deer mouse!), and so far as I know there's never been a case in a domestic rat.
The reason wild rats can be a health-risk to humans is not because they are "dirty and diseased", but because they are very like humans in their metabolism and diet, and so get a lot of the same diseases. A feral cat may have fifteen infections of which you might be able to catch one if you are very unlucky: a wild rat may have three infections and you can catch two of them.
By the same token rats can catch a lot of our diseases: don't breath on your rats or kiss them if you yourself have an infection and you don't know for certain that rats can't get it. You can certainly pass on the common cold to rats: although not, so far as I know, 'flu' (ferrets, however, can catch 'flu' from humans).
One thing domestic rats do get, and which they might be able to pass on to you, is ringworm, which they may arrive with or catch from e.g. hay at any time. Watch out for scabs forming at the corners of the rat's mouth and under the chin. If you see any, treat as described in the section on infection, and don't cuddle the rat too closely until the problem has cleared up.
Ringworm in humans usually manifests as a roundish red patch of little blisters. If you do catch ringworm from your rat, don't panic - any decent chemist will sell you an ointment which will clear it up in about 2 weeks. Nor is this problem specific to rats: most mammals can get ringworm, and a lot of them can pass it on (I once caught it from a rabbit).
Make sure wild rats or mice cannot get at the stored grain you use for your domestic rats - not only because they may spread infections, but because they may eat rat-poison and then pass it on in their urine.