Papayas have several medical properties, and native inhabitants of South America have used them for centuries in ethnomedicine to treat several diseases and symptoms.
Mature, ripe fruits have been used for centuries as an effective remedy against ringworm, due to its antielmintic properties.
Green fruits, on the other hand, have been used to lower blood pressure, and as an aphrodisiac.
Quite surprisingly, unripe fruits were also used to promote contraception and abortion, and this effect has been confirmed by modern research: as such it is now advised that pregnant women and couples refrain from consuming large amounts of green papayas before and during the first stages of pregnancy.
Due to papayas' high content in proteolytic enzymes, especially papain, their juice has been used to pre-digest meat (e.g. as a meat tenderizer), and it seems to be effective in reducing gastrointestinal gas. It is especially useful for those who suffer from IBS, since it reduces inflammatory responses.
Papaya seeds are used as a substitute for black pepper, due to their peppery flavor, but they also have important medical properties:

In folk medicine, they were used to reduce inflammation and pain due to their analgesic properties. Other uses include the treatment of stomachache and fungal infections, as well as worm parasitic infections (the seeds have anthelmintic properties).
Like green fruits, the seeds have been proven by research to possess strong spermicidal effects, and may temporarily reduce fertility or be used as a contraceptive (don't count on its effectiveness though).
Green leaves are used in several regions of Asia to make a tea believed to have protective properties against malaria. This effect has not been proven by scientific trials as of 2009. Leaves are also steamed and eaten like a vegetable, and are used as a heart tonic, analgesic and to treat stomachache.
Even the roots of the plant appear to have antalgic effects, and were sometimes essiccated and eaten raw or in teas to reduce inflammatory pain.