Wart removal surgery is always conducted at a hospital or clinic by a medical professional such as a doctor or dermatologist. You should never try to cut out a wart by yourself, since this can cause unnecessary pain and infection.
The main type of wart removal surgery is called curettage, but it is not uncommon for this procedure to be done in combination with another wart treatment, such as laser wart removal or electrosurgery/burning off warts. Your best option will depend on the location, size and type of wart. Curettage has been in use as a surgical method of treating warts for a relatively long time and it is still widely practiced. It is fairly quick and has a good initial success rate, although recurrence is not uncommon. However, it can be perceived as painful and cause scarring. It can also be quite expensive.