Even if you're not religious I still think you should take the time to visit churches or different faiths and see the place of worship - it's always interesting, especially when you're abroad as faith means different things to different cultures. Obviously Judaism is very prominent in Budapest/Hungary and the Great Synagogue is a beautiful building in the Jewish Quarter that shouldn't be missed. Be sure to time your visit as it closes in the early afternoon, we visited on Sunday then went for a Jewish lunch buffet at a nearby restaurant.
You need to buy tickets to enter the grounds, you can also do a guided tour - they did them in many different languages. Once you've paid you will go through a strict bag check/security gate. When inside you'll pass through a mini-souvenir shop (buy a fridge magnet!) then you can pass through to view the Jewish cemetery and into the main courtyard. There is another memorial wall and a beautiful metal tree with names engraved on each leaf. The actual synagogue itself is very pretty and once inside men must wear a kippur (provided) and you can wander in between the benches and view the details decor. You can go into another section of the synagogue which houses the small museum. A history of Judaism is displayed going up the staircase and inside there are some items and artefacts to view as well as a Hungarian language video clip.
Overall it was an interesting trip and I'm glad we went. I learnt a lot more about Judaism and the persecution of the Jews throughout history. There are some very good Jewish restaurants nearby which you can visit for refreshments - Carmel on Kazinczy St is definitely worth a try and does fantastic Flodni cake!
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