A fiber grinding process includes: contacting an end of the optical fiber with a grinding surface in such a manner that a central axis of the optical fiber at the end is inclined to the grinding surface; rotating the optical fiber about the central axis; and changing a contact pressure between the end of the optical fiber and the grinding surface by applying a variable torque while rotating the optical fiber. A substantially cone-shaped end face with a substantially elliptic cross-section may be produced. A fiber grinding apparatus is also disclosed.