These Squares Are Not Quite Square Quiz

Answer: Handschoenmarkt

The Handschoenmarkt is what we're looking for. It's a rather small triangular square between the Grote Markt (with the city hall and the statue of Brabo) and the Groenplaats (especially known for the statue of Rubens).
The Handschoenmarkt and surrounding streets and squares offer the tourist a great choice of restaurants and pubs. One of the most famous pubs, the Paters Vaetje (with 100 different beers) is almost on the Handschoenmarkt.
The Vrijdagmarkt (Friday Market) is a square where each Friday a flea market is held. On the Graanmarkt the new city theatre is established, and each Sunday there is a kind of a market. The Eiermarkt (Egg Market) houses a tiny statue of Teun de Eierboer (Teun the egg vendor).
Maybe I shouldn't have translated these street names. True Antverpians hate to have their street names translated, because the French authorities during the Napoleonic era made several false translations.

Answer: Gate of Heavenly Peace

Tienanmen Square is of course named after Tienanmen Gate, the first of three gateways from the Square to the Forbidden City. The phrase Tienanmen consists of the characters for peace, for heaven and for gate, so the most obvious translation is "Gate of Heavenly Peace". Reality is of course a bit more complex, as the Manchu transcription shows. The Manchu text is equivalent to "Gate of Heavenly Peacemaking", but the longer Chinese definition is "receiving the mandate from heaven and stabilizing the dynasty".
In contrast with this peaceful name, I think most of us will remember the Tienanmen Square for the iconic picture of a lone demonstrator standing before a column of tanks, ready to move.
The Tienanmen Square is situated between the Gate of China in the south and Tienanmen Gate in the north. The east side is occupied by the National Museum of China and the west side by the Great Hall of the People.
When passing through Tienanmen to the north, one crosses the Upright Gate and then the Meridian Gate before entering the Forbidden City.

Answer: Place de la Concorde

The Place de la Concorde is indeed the start of the Champs Elysées, or the end if we travel from the Place Charles de Gaulle. The square was designed as an octagon (but not a regular octagon, more of a rectangle with cut-off corners), but it contains an elliptic driveway. In the middle of the shape we find an Egyptian obelisk transported from Luxor. At equal distances north and south from the obelisk there is a fountain: the Fontaine des Mers (Fountain of Seas) in the south, and the Fontaine des Fleuves (Fountain of Rivers) in the north. To the west is the avenue des Champs Elysées, and to the east is the garden of the Tuileries.

The Place de la Concorde (Square of Concord) was previously known as the Place Louis XV (up to 1789) and the Place de la Révolution (up till 1795). Most of the public executions by guillotine during the French Revolution took place on this square. After Napoleon was defeated, the square was renamed Place Louis XV or even Place Louis XVI, before receiving its name Place de la Concorde in 1830.

The Place des Pyramides, next to the rue de Rivoli, contains not a single artifact that refers to Egypt. Instead, we find an equestrian statue of Joan of Arc.

Place Pigalle is a semi-circular square at the foot of Montmartre, where a tourist might end up after descending from the church of the Sacré Coeur on top of the hill.

Place de la Madeleine is the site where we find the church of La Madeleine (previously a Roman basilica or market temple).

Answer: Tahrir Square

Tahrir Square (meaning Liberty Square) was originally named Ismailia Square, after the nineteenth-century khedive Ismail. The main artwork on the square is a statue of the Egyptian nationalist Omar Makram right in front of the Omar Makram Mosque. The Egyptian Museum borders Tahrir Square, as does the Mogamma government building.
The demonstrators in 2011 gained their cause: President Mubarak was deposed and a general election in December 2011 cleared the path for President Morsi. Howver, by the end of June 2013, the majority of the Egyptian people were dissatisfied with the way Morsi handled various problems (economical issues, border security, plans to reform the judicial structure, a highly disputed constitution etc.) General Abdul Fatah al-Sisi led a coup and appointed chief justice Adly Mansour as interim president. Morsi was given house arrest.

Taksim Square is the main site of anti-government protest in Ankara, Turkey, in 2013.

Medan Merdeka (Independence Square) is situated in Jakarta, Indonesia.

Naghsh-e-Jahan is the former name of Imam Square in the Iranian city Isfahan. The name Naghsh-e-Jahan translates to "Image of the World".

Answer: Sergels Torg

Sergels Torg was named after the sculptor Johan Tobias Sergel, whose workshop during the 18th century was just north of this square in Stockholm. To the northeast we find a roundabout with Sergels Fountain in the midst. The pedestrian area in the west is paved with triangular pavements, alternating between white and gray. To the south we find the Kulturhuset (Culture House) and the Stockholm city theatre.
Pilsudski Square, another trapezoid, is situated in Warsaw. It was named after the Polish head of state between 1918 and 1935.
Mzilikazi Square, yet another trapezoid, is the main square in the eponymous suburb of Bulawayo (Zimbabwe). Both the suburb and the square were named after the king of Matabeleland until 1868.
The Red Square in Moscow is situated before the Kremlin complex. One of the main buildings on the square itself is the mausoleum for Lenin.

Answer: Aotea Square

Aotea Square is the only one of these four squares situated in New Zealand, the rest are in Europe. Wellesley Street to the north is almost a straight line, and Mayoral Drive is a curve that together with Wellesley Street makes a ground plan resembling a human heart. Midway through this ground plan runs Queen Street. One half of the ground plan is formed by the Auckland University of Technology (main building between Rutland Street and Wakefield Street, library between Rutland Street, Lorne Street and Mayoral Drive), while the other half is the aforesaid Aotea Square. The Auckland Town Hall, the Auckland Civic Theatre and the Aotea Centre (also a theatre) are built on the Aotea Square.

Orlov Most is a bridge and square in Sofia, Bulgaria, Ban Jelacic Square is in Zagreb, Croatia, and Kossuth Lajos Square is in Budapest, Hungary.

Answer: Athens

You may have derived from the original name that this square is somewhere in Greece. Alas, the red herrings are also Greek cities with important squares.
Syntagma Square is in the capital Athens, between the Vasileos Georgiou Street, Othoonos Street, Amalias Avenue and Filellinon Street. Its curious shape is caused by the curve that the Filellinon Street makes. Across Amalias Avenue you'll find the Greek Parliament.
Between 2010 and 2013, the gloomy economic situation and very drastic economic plan instilled into many people the desire to protest in front of the Parliament. So they met on Syntagma Square, where they clashed with the police and army guarding the Parliament buildings.

The Plateia Georgiou I in Patras is another interesting square, shaped as a wristwatch.

Thessaloniki's major square is Aristotelous Square, next to the harbour.

Mytilene holds the Sappho Square, a trapezoid square named after the most famous inhabitant of Lesbos, the ancient poetess Sappho.

Answer: Mothers of the Plaza de Mayo

The Plaza de Mayo was named after the month in which the 1810 revolution started, which led to the independence of Argentina. The Plaza hosts an obelisk erected to commemorate this revolution.
Around the Plaza de Mayo, we find the Casa Rosada (Pink House: the seat of the government) as well as the city hall, the Metropolitan Cathedral and the Bank of the Argentine Nation.
The Mothers of the Plaza de Mayo were relatives (mainly mothers, but also others) who gathered on the Plaza each week, with pictures of their relatives disappeared (presumably by order of the government). During the reign of president Videla, a right wing politician, many people who uttered leftish messages suddenly "disappeared". In later years the truth came out: these young people with leftist sympathies, were abducted by the army or by paramilitary militias, then tortured and finally killed and dumped in desert areas.
I've invented the other movements, but not the respective squares. Bubenbergplatz is in Bern (Switzerland), Place d'Youville is in Montreal and Quebec (both in Canada), and Rizal Park is in Manila (Philippines).

Answer: Gian Lorenzo Bernini

Bernini (1598-1680) is best known for his sculptures, many of which decorate Rome and Vatican City, but he also painted and worked as an architect. Some of his most famous buildings are the Palazzo Barberini and the Palazzo Chigi in Rome, besides the aforesaid St. Peter's Square. His sculptures include (among others) a David, the Ecstasy of Saint Theresa, Saint Peter's Chair (and baldachin), the Fountain of the Triton and the Fontana dei Quattro Fiumi.

Brunelleschi (1377-1446) is best known for the cupola on Firenze's Duomo.

Michelangelo (1475-1564) worked as a sculptor (David, Pieta) and as a painter (the ceiling of the Sistine Chapel).

Vasari (1511-1574) worked principally in Florence. He renovated the Santa Maria Novella and designed the loggia of the Uffizi. His best known contribution to art is the biography of Renaissance artists.

Answer: Civic Square

To the north of Civic Square, we find Harris Street. To the east is Jervois Quay and the harbour. To the west is Victoria Street, which curves into Wakefield Street to the south. It's this curving that explains the peculiar shape of the square. The Civic Square was first planned in 1944, but work began only in 1977. By 1992 it was completed.

ANZAC Square is in Brisbane, Australia.

Lemon Quay is a small square in Truro, England.

Times Square is the well known square in New York City.