Bebelplatz is surrounded by the State Opera (Staatoper), St. Hedwig’s cathedral and the Alte Bibliothek (the old library), and is located across the road from the Humboldt University.
The State Opera House was built in the 1740s, but was completely destroyed by fire in 1843, and rebuilt by Langhans. It was modernized in 1986, and is now led by the esteemed conductor Daniel Barenboim, an Israeli by nationality, although a pretty controversial figure in Israel. I myself adore him, but people here decided for some reason to boycott Wagner (who died in 1883, btw), because Hitler liked him (I wonder why they don’t boycott vegetarianism as well), so his insistence on Wagner’s cultural importance doesn’t go so well here. And also his love for peace.
St. Hedwig’s Cathedral was built in 1747, as a symbol of the religious tolerance of Prussia and Friedrich the Great. It was the first Catholic Church constructed in Germany after the Protestant Reformation, and is still the most important Catholic Church in Berlin. It was modeled after the Pantheon in Rome, and its construction was financed by donations from all over Europe.
It was also one of the centers of Catholic resistance to the Nazis, under the leadership of Bernhard Lichtenberg, who died in prison (before being deported to Dachau concentration camp). The church was badly damaged in WW2, and rebuilt in the 1950s.
This is the official site of St. Hedwig Cathedral (you can find there opening hours and schedule of masses and organ concerts).
Alte Bibliothek, or The Old Library, is located on the west side of Bebelplatz, and is now a part of the Humboldt University. The rest of the campus is across the street.
Humboldt University (Unter den Linden 6) was established in 1810 by Wilhelm von Humboldt, and was then called “Friedrich Wilhelm University”. It was dedicated to the spirit of enlightenment, and many “big names” either studied or taught there. Among the students were Karl Marx and Friedrich Engels, Heinrich Heine and the Grimm brothers (from the children’s tales). Among the famous teachers were Albert Einstein, Hegel and Max Planck. The university was named after its founding father in 1949.
The equestrian statue of the Prussian King Friedrich the Great stands in front of the university. The statue was created by Christian Daniel Rauch and unveiled in 1851.