Although there is no set age, it is best to wait until your breasts have finished developing. You can better determine this with your OBGYN if you are a longstanding patient of his or hers. This can vary and although you may think you have finished maturing by 18 or 19 - your breasts will continue to go through changes well into your early twenties.
The youngest is usually 18 although in special cases of pronounced asymmetry and reconstruction - prostheses can be used on persons younger than 18. These younger cases are very specific.
On the other side of the spectrum, women in their late 60's who have gotten breast implants. When we are older the only thing that may stop us from having breast augmentation surgery is general health so be sure to have a physical to see if you are in good health to properly heal and handle the anesthesia factor.