The Babylonians, Ancient Greeks & Romans and Egyptians all used to make soap by mixing fat, oils and salts but soap they they didn't make soap for bathing and personal hygiene but for cleaning cooking utensils or for medicinal purposes.
The Babylonians made soap from fats boiled with ashes but the Egyptians used vegetable and animal fats with alkaline salts to produce a soap like substance. The early Romans made soap from urine!
More recently (although still thousands of years back) soap was made by mixing animal fats with lye, which is still a very common practice nowadays. Some butchers still keep the fat from the slaughter of animals to sell to soap makers to make soap, ensuring every part of the animal is used.
Modern soap making has pretty much stayed the same since around the 1800's where it was identified that there was a relationship between glycerin, fats and alkaline. During the mid-nineteenth century soap for bathing become a separate commodity from laundry soap and more mild bars were made for application to the skin. Liquid hand soap was invented in the 1970s and since then it has been the preference but with the drive for more eco-friendly products the bar soap is making a come back.