Franziskanerplatz is one of these places I spotted on the Internet even before I came to Vienna and I was lucky because I used to live very close to this lovely square which has a fountain in the middle and looks like one of those squares somewhere in Italy. Unfortunately my flat now is not so close to Franziskanerplatz but when the weather is nice in Vienna, visit the Kleines Café in Franziskanerplatz (they serve delicious bread with a choice of toppings) and wait for a while until a table is free outside. I never visit this place during the winter because the cigarette smoke indoors disturb me a lot, but during the summer is definitely a very pleasant place, less touristy, central yet quiet.
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Franziskanerplatz 3, 1010 Vienna, Austria