Vienna is a party to the Schengen agreement. As such, U.S. citizens may enter Vienna for up to 90 days for tourist purpose without a visa. The passport should be valid for at least three months beyond the period of stay.
American citizens traveling for tourism is not required to have a visa for this initial entry into the Schengen area, but must have a passport valid three months beyond the proposed stay (for a two month holiday the passport must be valid for five months, for example.)
Please note that non-U.S. passport holders are responsible for obtaining the required documentation applicable for their entry.