The type of the anesthesia varies with each procedure. For some procedures (e.g. facelift procedures), only local anesthesia are used. For liposuction, a local anesthesia plus oral sedation is typically recommended. This method will allow you to be awake, but insensitive to any pain. In more extensive procedures, such as a full abdominoplasty (tummy-tuck), or complex revision procedures, general anesthesia is used so you will be asleep during the entire procedure.
The specific type of anesthesia used in your operation will depend on the type and extent of the procedure(s), your medical condition, and the recommendation of your surgeon.