You do not need a permit to build an at or below 8' high fence of any material, i.e. wood, metal or stone provided it is not located within a floodplain hazard area. If you are uncertain whether your property is located within a floodplain area, please contact the Watershed Protection Department at 512-974-2843.
You can build a "solid" fence between the heights of 6 and 8 foot, given it either meets the requirements of City Code 25-2-899 (Sections E-G) or you obtain a variance from the City of Austin Board of Adjustment. In both circumstances the homeowner will be required to obtain written authorization from the adjacent property owners that are affected by the fence. If the adjacent property owner is the City of Austin, you will NOT be able to obtain that authorization letter; you must go straight to the Board of Adjustments and ask for a variance.
You cannot build a "solid" fence over 8 feet high. A fence considered "ornamental" may be built higher than 8 feet after a permit has been obtained. Please see Section (B) of City Code 25-2-899 for the difference between "ornamental" and "solid."
The City of Austin does not regulate, delegate, prevent or police the types of materials that may be used in the construction of a fence.