The minerals in the water are said to be good for arthritic conditions, and you can watch bathers playing chess while enjoying the soothing heat in one of the outdoor pools.
It would be easy to spend a whole day there, and many do, but there are things to see, places to visit.
One fact that most people know about Budapest is that it is really two cities - Buda on the hill on the west of the broad Danube, and Pest on the flat eastern bank.
Buda is the old city, with quaint colourful houses, and Pest is largely a creation of the mid-19th to early-20th centuries, all broad boulevards, grand mansions, neo-Baroque and art nouveau.
So what to see in a short stay? Budapest is certainly a city of museums. Whatever your interest - even if it's telephones or marzipan - there's a museum to appeal to you.
We chose just two, both housed in imposing mansions and, conveniently, almost opposite each other on the grand Andrassy Ut boulevard, a World Heritage site in itself.
The first was the Franz Liszt museum, in the apartment where the great composer spent the last years of his life. In the room where he slept and entertained, you can see the desk on which he composed, complete with a clever little pull-out keyboard. In another room is the piano he played, and another on which he taught.
Over the road, number 60 houses a different sort of museum entirely - the House of Terror. This was the HQ of the notorious Arrow Cross, the Hungarian fascist party, and later, after the war, of the communists' feared state security organisations.
The interior has been transformed into a breathtaking and very modern museum of totalitarianism, and it saves the best - or worst - until last. Most of the exhibits are on the upper floors and a lift takes you back down afterwards. But this is a very slow lift, and as it descends to the basement, a former official describes on video the horrific way in which prisoners used to be executed.
When the lift finally arrives in the basement, you're left to wander around the one section of the museum that needs no explanatory captions... the cells where prisoners were held and tortured, and the gallows where they were executed.
It is a sombre experience - but Budapest is a wonderful destination for the melancholy tourist.
History weighs heavily on the city.
"The last hundred years have been turbulent for Budapest," I suggested to one local.
"One hundred years?" she responded.
"Many, many centuries."
Budapest was restored after the battering it received at the end of the Second World War - it is the only city where you can still see the wartime Jewish ghetto - and is a welcoming place these days, even if you're not a melancholy tourist.
You could just go there for the food and drink. The beer and wine are excellent, and you'll love the city if you like cake.
Goulash soup is on every menu, but Hungary has a rich cuisine, making much use of pork, beef and veal.
One evening, my wife and I ate in a cosy little restaurant called Spinoza in the old Jewish quarter, where we made sure we tried different dishes. Both were based on smoked goose - very Hungarian.
For a cheap, filling lunch, go to the massive Great Market Hall, designed by Gustave Eiffel, of Tower fame, where upstairs stalls sell hearty food at great prices. We shared a plate of pork and potatoes for just 1000 forints (about NZ$5.40) and two large beers for the same again.
Since the John le Carre days of the Cold War ended, all the big international hotel chains have moved in, but you can still stay in places with character.
Our hotel was the Danubius Astoria, built 100 years ago in the dying years of the Austro-Hungarian empire and lavishly decorated in marble, gilt and mirrors. It's an old-style grand hotel of a type that is hard to find these days, complete with a porter who materialises when the receptionist pings the bell.
The hotel has been at the centre of modern Hungarian history, playing a role in the Hungarian independence of 1918, the Nazi occupation and the Soviet crushing of the 1956 uprising.
But if you find all that history hard to live with, you might enjoy the quirky and very modern riverside art'otel. American artist Donald Sultan not only provided all the pictures on the walls, but also wittily designed everything from the carpets to the bedding.
English might not be quite as widely spoken as in many countries, but the ladies in the Metro station will know what you mean when you say "24-hour ticket". And you should, as this 1550 forint bargain will take you all over the city.
Ride the bus up to the picturesque old town in Buda, travel the trams along the Danube and around the great boulevards, and make sure you use Metro Line 1, which locals will proudly tell you is the oldest underground line in Europe - after London.
It has beautiful tiled stations, rickety old trains - and it travels all the way under Andrassy Ut to the Széchenyi baths.
IF YOU GO
Best for: Plenty of history.
When to go: The city can be very cold in winter and hot in summer, so spring and autumn are recommended.
Don't miss: A visit to one of the spas, obviously - but also a cake at the historic and very fancy Cafe Gerbeaud.
Need to know: Say cheers but don't clink glasses - Austrians did it after executing Hungarian generals in 1849 and apparently it can still cause offence. That's how long memories are in Budapest.
Don't forget: Your swimming costume.
- PAA
