Rudkøbing was the seat of the former Rudkøbing Municipality (in Funen County), and is the seat of the current Langeland Municipality (in the Southern Denmark Region). When Rudkøbing was its own municipality, there were often negotiations with the two other Langelandic municipalities of Sydlangeland and Tranekær to merge together. These plans were repeatedly abandoned due to a lack of public interest. In the Structure Reform of 2007, the three were forced together, and Rudkøbing has since been the municipal seat.
The Langeland Bridge connects Rudkøbing (and the rest of Langeland) to the island of Siø. Siø is further linked by the Siøsund Bridge to Tåsinge, and Tåsinge is linked by the Svendborgsund Bridge to Svendborg on the mainland of Funen, making Rudkøbing accessible by road.
Each year, the town hosts Langelandsfestivalen, a music festival which targets families and older people (unlike the larger Roskilde Festival elsewhere in Denmark, which is mostly attended by young people).
The first mention of Rudkøbing was in 1287, when it got market town privileges, though the town presumably existed before then. During the Count's Feud (1534-1536) and again during the Dano-Swedish War of 1658-1660, Rudkøbing was under siege by, and eventually had to give in to, Swedish troops. The town was hit by the plague during the 16th and 17th centuries, and experienced fires in 1590 and 1610.
The Ørsted family, which most notable includes the brothers Hans Christian and Anders Sandøe Ørsted, has its origins in Rudkøbing. Their father was a pharmacist, and the brothers were born in the old pharmacy on Gåsetorvet. Across from their birth house is a statue of Hans Christian, and a statue of Anders is located at nearby Ørstedsparken, which is named in their honor. The old pharmacy remained in use until 2017.
Rudkøbing got its first proper harbor in 1826 (it had previously only had a pier). In 1898, in a town of about 3,500 inhabitants, there were 152 ships stationed in Rudkøbing. Over 2,500 ships (carrying over 18,000 metric tons of cargo) came in and over 2,800 ships (carrying over 11,500 metric tons of cargo) went out every year, mostly from elsewhere in Denmark. The town was connected by steam ferry routes to Copenhagen, Korsør, Marstal, and Svendborg.
From 1911 to 1962, Rudkøbing was the main station of the Langeland rail network. The network was connected to Svendborg in 1926, when a railway ferry route made it possible to move train wagons between Rudkøbing and the mainland of Funen.
The town's population was stable during the Interwar period, hovering above 4,000, with the census in 1930 showing a population of 4,129. The population peaked at 4,541 in 1950, and by 1965, it had fallen to 4,204. Around this time, a suburb developed in Skrøbelev Parish, which eventually grew together with Rudkøbing.
Bridges connecting Rudkøbing and Svendborg were built throughout the late 1950s and early 1960s, finishing in 1966. The old ferry route between the two towns gradually lost popularity due to the bridges, and the route was eventually ended. The ferry route to Marstal, which had been active since 1866, was closed in 2013, so that the only ferry on this route could be moved to the busy route between Marstal and Svendborg. People in Rudkøbing wishing to get to Marstal now have to pass through Svendborg, rather than go directly to Marstal.
- Gertrud Skomagers (died 1556), who was accused by Hans Ipsen in Rudkøbing of being a witch. The case against her contributed to a change in the law regarding witch trials in Denmark
- Erik Bredal (1608 in Rudkøbing – 1672), a Danish-born Norwegian Lutheran bishop
- Otto Fabricius (1744 in Rudkøbing – 1822), a Danish missionary, naturalist, ethnographer and explorer of Greenland
- Hans Christian Ørsted (1777–1851), a physicist and chemist who discovered that electric currents create magnetic fields, known for Oersted's law
- Anders Sandøe Ørsted (1778 in Rudkøbing – 1860), a politician and jurist; Prime Minister of Denmark 1853-1854.
- Mads Johansen Lange (1807 in Rudkøbing – 1856 in Kuta, Bali, Indonesia), nicknamed the King of Bali, a Danish trader, entrepreneur, and peace maker on Bali. Several descendants, including Crown Prince Tunku Abu Bakar of Johor, visited Rudkøbing in 2011. 
- Anders Sandøe Ørsted (1816–1872), a botanist, mycologist, zoologist and marine biologist
- Jens Christian Bay (1871–1962), a Danish American writer and librarian, who emigrated to the USA in 1892
- Ingrid Larsen (1912–1997), a diver who competed in the 1932 Summer Olympics
- Preben Lerdorff Rye (1917–1995), a film actor who appeared in 92 films between 1941 and 1989. Appearances include the film Babette's Feast, which won the Oscar for Best Foreign Language Film
- Bjørn Watt Boolsen (1923–1998), actor, who played several characters in the Olsen-banden films
- Hans Jørgen Boye (born 1942), a retired rower, competed in the coxless pairs in the 1964 Summer Olympics
- Erik Larsen, (born 1944) politician and mayor of Egebjerg Municipality 1978-1986, member of the Folketing 1987-2005, for the Liberal Party.
- Nikolaj Coster-Waldau (born 1970), actor, who plays Jaime Lannister in the HBO series Game of Thrones
Closeup of the Hans Christian Ørsted statue on Gåsetorvet.
The Langeland Bridge, as seen from Rudkøbing.
Gåsetorvet seen from behind the Ørsted statue. The building with the staircase is the old pharmacy.