Electrolysis

Electrolysis is your best bet for permanent hair removal. In electrolysis a low electrical current is sent to the hair root via a hair-thin needle, effectively 'killing' the hair and root for good.
Because every hair must be treated one at a time, electrolysis takes a long time. Multiple treatments are required.
Electrolysis can be performed most anywhere on the body, including the eyebrows, face, legs, genitals, bikini area, and back. Though sometimes a physician's consultation is required for removing hair from areas such as moles.

ELECTROLYSIS - HOW TO DO IT
First of all, there are "home electrolysis" kits which are questionable and not recommended.
Professional electrolysis is highly likely to be more effective and less risky.
To find a reputable elecrologist, first of all look for one that is licensed or certified with the state, if your state requires it. Make sure the certificate is on display and current.
If your state does not require them be to certified, try to find one with a certificate from an accredited electrology school.
Most places will offer a free consultation, which is a good way to get an idea of their particular procedure. During this you can ask any questions you may have regarding cost, how much discomfort to expect, how many treatments you may need, etc.
Finally, make sure the place looks clean. Do they use disposable gloves and probes? Does the place look sterile like a doctor's office, or shady like a back ally tattoo parlor? Ask to meet the electrologist. Do they seem professional? Finding a place that you're comfortable with is important.

ELECTROLYSIS VERSUS OTHER METHODS
Compared to other hair removal methods electrolysis is rather costly and painstaking. However, it is your best bet for permanent hair removal. Laser hair removal is less painful, expensive, and time consuming, however the hair removal is not as extensive or permanent.
And unlike shaving, waxing, and other temporary methods, once treatment is over there is no further cost or upkeep.
Basically if you want to remove a small amount of hair from a problem area such as the upper lip or breasts, electrolysis is a good bet. However larger areas such as the legs or chest will be a much longer and costly process, and laser hair removal may be the more appropriate choice unless you are seriously devoted to electrolysis.