I am American. My Italian girlfriend who lives in Vienna says that I may have to apply for an ESTA visa to visit her. Is this correct? The last time I went to Europe (not Austria) I did not have to fill out a visa...
The answer is indeed no; no visa or ESTA of any kind needed for a visit of 90 days or less.
Here's what the US State Department (Bureau of Consular Affairs) has to say:
ENTRY / EXIT REQUIREMENTS FOR U.S. CITIZENS:
Austria is a party to the Schengen Agreement. As such, U.S. citizens may enter Austria for up to 90 days in any 180-day period for tourist or business purposes without a visa. The passport used should be valid at least for the period of the intended stay (usually the date of the return flight).
Anyone intending to stay longer than 90 days must obtain the appropriate visa issued by the Austrian Embassy in the United States. Specifically students attending a college/university in Austria should consult with the Austrian Embassy/Consulate before coming to Austria. Information for students is available from the Austrian agency for international mobility and cooperation in education, science and research (OeAD).