Budapest, Hungary was where we spent the most time on our trip in October, 2009; we were there about a week over two separate visits. It's a very thriving city which is very easy to get around, both by transit and walking.
Budpest is divided by the Danube (Duna) river into the hilly Buda on the west and the flat Pest on the east.
Parliament Building (Országház)
One look at the outside of Hungary's Parliament building tells you that it was meant to impress. It has literally miles of stairs, and currently is much larger than it needs to be. The House of Commons meets on the south side (left in the picture), while the House of Lords used to meet on the north side. The latter is now sometimes leased out for events.
Hungarian State Opera House (Magyar Állami Operaház)
The Opera House was celbrating its 125th year when we were there. it was built not long after Budapest became co-captial of the Hapsburg Empire. Half of the funds were provided by Emperor Franz Josef on the condition that it be smaller than the Vienna Opera House, which it is. However, this opera house surpasses the one in Vienna in opulence.
City Park (Városliget)
The 1896 Millennium Exhibition celebrating Hungary's 1,000th anniversary was held in City Park, and it included Vajdahunyad Castle (Vajdahunyad Vára). It was built with temporary materials, and because locals liked it so much the castle was rebuilt with brick and stone after the exhibition. It has a mix of styles; the left picture shows the Gothic gate, and on the right is a replica of a Renaissance Transylvanian castle. The right picture shows the Romanesque Benedictine chapel on the left and a Baroque mansion, home of the Museum of Hungarian Agriculture (Magyar mezőgazdasági Múzeum).
St. István's Basilica (Szent István Bazilika)
St. István's Basilica was also built around the time of the 1896 Millennium Exhibition. Its design took over 50 years and involved three architects, each who favored different styles. This is the largest church in Budapest.
Hungarian National Gallery (Magyar Nemzeti Galéria)
The National Gallery is on Castle Hill on the Buda side of the Danube. It's part of the Royal Palace (Királyi Palota), once considered the top Renaissance palace in Europe.
Castle Hill (Várhegy)
Other sights are on Castle Hill in addition to the National Gallery. This column from 1713 was erected by survivors of the great plague. It's in the Holy Trinity Square (Szentháromság tér).
Many communist monuments were removed from Budapest after the fall of communism. Some of them were re-erected at Memento Park (Szoborpark).