At what age do you reach your permanent height?

Depends. Final height is reached when the growth centers of your bones close and further linear growth cannot occur. This varies in most men from 16-20 and women from 15-17. A single x-ray of the wrist/hand can be assessed for open growth centers. When open, more growth is possible, if closed it is finished.
Depends on gender. Women reach their full height about two years after menarche (the first period). Men may continue to grow until 20 yrs of age.
16-19. After your growth plates close you stop growing in height. This is typically somewhere between 15-17 years of age for girls and 17-19 years of age for boys.