There are a few things to watch out for as far as your rat's health is concerned. I am not a vet and do not profess to be an expert on the subject but I will explain to the best of my abilities.
Greasy Patches;
This is usually found in mate rats and is caused by too much oil being produced by the oil glands. Bathing will help the appearance but if the problem is due to over production and not a one time occurrence the problem will re-occur. It does not harm the animal but it is rather unsightly.
Hair Loss.
If your rat is losing hair there are several possibilities. Mutual grooming from two or more rats will sometimes result in the loss of hair from some areas of the body. Also a red staining may occur from this as well. This is because rat's saliva has a red colouring agent in it and may transfer to the coat. This will especially be apparent in light coloured rats.
If the rat is suffering from mites or lice the animal may be scratching or rubbing the hair off. Mites and lice need to be tended to. Mites are usually visible around the nose, ears, and tail. Lice themselves are not usually seen but their e s can be seen attached to the hair shafts. If you blow on your rats fur and little "dust particles" do not blow off, they are lice eggs. For both of these problems, the rat, housing and furniture (toys, dishes, water bottle) must be thoroughly cleaned. Scrub the cage/aquarium, dishes, toys and water bottle with a mild bleach solution and rinse well. The rat must then be bathed before being placed back into the clean cage. Use a mild cat flea/tick shampoo and bathe the rat gently but thoroughly using a cotton ball around the eyes and head. I have had good success with a bird lice/mite spray. Make sure it does not get in the animals eyes or nose. I find this works well if your rat is particularly afraid of water. (some rats love a bath and will even enjoy a swim!) This can also be sprayed on the cage before re-bedding it to be sure of the mite/lice removal.
If your rat is carrying the rex or hairless gene in it's background it may also lose hair which may or may not grow back.
Lastly an allergy may be the culprit. Rule out everything else before going to this, as this is usually rare. (Make sure you are NOT using cedar bedding!!)
Eye Infections.
These are not common problems but do occur. Usually the culprit is a toenail in the eye or dust/shavings. Clean out the eye with a soft cloth and warm water. Apply a veterinary antibiotic ointment. I like to use "Polycidin" It is a human eye/ear antibiotic drop. I have used it successfully in mild cases on rats, guinea pigs, rabbits and birds. If you see a greyish white area on the eye this is a sign of cataracts and there is no treatment. If the eye seems to be cut or protruding take the rat to a vet immediately. This could be serious and not to be treated at home.
Vaginal Bleeding.
Rats do not bleed during their heat cycles. If your rat is pregnant it may be the signs of labour, miscarriage or labour distress. If the rat is not pregnant this is a sign of uterine cancer and should be seen by a vet to determine the extent of the cancer and possible treatment options.
Wounds,
Sometimes wounds occur from fighting between rats or accidents. Inspect the wound - if you can see tissue that is off-white to grey in colour - the wound is serious and requires stitches. If it is superficial, clean the wound with warm water and a soft cloth. Then mix equal parts hydrogen peroxide and water and apply to the wound. The bubbling action of the peroxide will cleanse the wound and kill any germs present. (2 people are recommended for this course of action, as the rat will not enjoy it!) Keep the rat calm and repeat the cleansing as needed.
Abscesses.
These start from an uncleaned or unnoticed wound. The wound closes over without being cleaned and bacteria as allowed to grow. The abscess must be opened, drained, and disinfected by your vet. This is necessary as I left unattended to, the poison could enter the bloodstream and our rat.
Tumors:
Unfortunately rats are very prone to tumors. This is why they have been used so extensively in medical research for years. Most rats will acquire tumors at some time during their life or at the onset of old age. They usually do not bother the rat, only us. Some types are operable Shoe others are not, please consult your veterinarian to recommend a course of action.
Spaying/Neutering
This is an operation, which removes the animals sex organs. While this is an option, it carries high risks due to the small size of your pet and the need to be put out under anaesthesia. Also it may prove difficult to find a vet comfortable with the procedure. The benefits would be the reduced risk of uterine, mammary, or ovarian cancer and will allow a male rat to live with a female without the risk of babies.