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We celebrated new years eve in Budapest last year. It was a very interesting experience. There is a lot to do in that period. I would suggest to be on one of the bridges at midnight to enjoy the fireworks. There are also different river cruises you can book so that you are actually on the Danube river at midnight. We opted out for the latter, since it would involve being on that boat for some time and we preferred to walk around. We were a large group, also with some small children and we never felt unsafe. We were however lucky to find accommodation right in the center of Budapest.
The most funny feature on Budapest during New Years' eve is that people wear wigs, funny faces and other disguises. In the days before New Year's eve you will see a lot of street vendors selling these wigs, horns, lights and other attributes.
I can't suggest a specific bar or disco, since I am not a local. However, the streets in the center have a great atmosphere and afterwards you just need to follow the crowds and just pick a place to you liking.
At midnight we were at this bridge and we had a stunning view of the city and the fireworks. Bring extra layers of clothing though, standing on a bridge above the Danube river can be rather cold at that time of the year.
Not related to New Years eve, but relevant to visit in that period is the SzΓ©chenyi Thermal Bath. They offer Disco on certain evenings of the week. It is really great to enjoy a thermal bath that way.
Together with New Years' even, the Nutcracker it formed the top three of the highlights during our trip.