There are many places throughout the city where you can listen to Gypsy music. Since most of these places are restaurants, and the programs are usually in the evening, you could also eat dinner or have a drink in the meantime.
First, and foremost (as an exception), there is the Napház Khamorro Minority Cultural Institute (Budapest VIII. 31 Népszínház street), which offers programs all day long, plus every night features a different aspect of Gypsy music. If you would like to get a deeper insight into the gypsy folklore and art as well, this is definitely the place to go.
There are many other places of interest though, most of them featuring a wide variety of music, from "mulatós" (a special kind of pseudo-folk music) to jazz. To experience music, gypsy food, and folklore, you should visit one of the inns and restaurants listed at the following page of Romaweb.hu. (http://www.romaweb.hu/romaweb/index.jsp?p=esem&sp=view&id=659) Despite of not being widely advertised, there are many places where you can get acquainted with gypsy culture in person.
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Traditional gypsy music in Budapest
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Popular cinema in Hungary
Hungarians learning Spanish
Hungarian attitudes towards Gypsies
Austria from Hungary
Gypsy stereotypes in Hungary
Getting a job in Hungary
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Foreign languages in Hungary
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Gypsies in Hungarian media