HPV and wart virus cannot be cured or even realistically prevented, but visible genital warts can be treated if necessary. (You can read about the available treatments for cervical cell changes due to HPV in pap test.)
The success of genital wart treatments varies and an individual person's response to treatments is unpredictable. As mentioned before, in many cases the person's immune system successfully overcomes the HPV virus and the physical signs disappear on their own. Sometimes it is hard to know if the treatments actually helped, or if the warts were going away regardless. However, other people find their warts grow and spread and/or continue to appear and reappear for years, despite repetitive treatments.
Many people request that their warts be treated because they look unsightly or are causing discomfort. Some people prefer to treat their warts early, in case they grow and spread, while others are happy to 'wait and see' if the warts go away on their own. The types of treatments used will depend on the size and location of the warts and your (and your caregiver's) preferences. However, even though the warts may be removed, the underlying viral infection is not cured. This is why warts often return. The different treatments available include:
Trichloroacetic acid
Podophyllin
Podophyllotoxin
5-Fluoruracil (Efudix, Fluoroplex)
Imiquimod (Aldara)
Interferon injections
Cryotherapy
Cautery or electrosurgery
Surgical removal
Laser therapy
Trichloroacetic acid.