Auctions have been recorded as early as 500 BC. According to Herodotus, in Babylon, auctions of women for marriage were held annually. The auctions began with the woman the auctioneer considered to be the most beautiful and progressed to the least beautiful. It was considered illegal to allow a daughter to be sold outside of the auction method. Attractive maidens were offered in a forward auction to determine the price to be paid by a swain, while unattractive maidens required a reverse auction to determine the price to be paid to a swain.

Auctions took place in Ancient Greece, other Hellenistic societies, and also in Rome. During the Roman Empire, after a military victory, Roman soldiers would often drive a spear into the ground around which the spoils of war were left, to be auctioned off. Slaves, often captured as the "spoils of war", were auctioned in the Forum under the sign of the spear, with the proceeds of sale going toward the war effort.

The Romans also used auctions to liquidate the assets of debtors whose property had been confiscated. For example, Marcus Aurelius sold household furniture to pay off debts, the sales lasting for months. One of the most significant historical auctions was in 193 AD, when the entire Roman Empire was put on the auction block by the Praetorian Guard. On 28 March 193, the Praetorian Guard first killed emperor Pertinax, then offered the empire to the highest bidder. Didius Julianus won the auction at the price of 6,250 drachmas per guard, an act that initiated a brief civil war. Didius was beheaded two months later when Septimius Severus conquered Rome.

From the end of the Roman Empire to the 18th century, auctions lost favor in Europe, while they had never been widespread in Asia. In China, the personal belongings of deceased Buddhist monks were sold at auction as early as the seventh century AD.
The first mention of "auction", according to the Oxford English Dictionary, appeared in 1595. In some parts of England during the 17th and 18th centuries, auctions by candle began to be used for the sale of goods and leaseholds. In a candle auction, the end of the auction was signaled by the expiration of a candle flame, which was intended to ensure that no one could know exactly when the auction would end and make a last-second bid. Sometimes, other unpredictable events, such as a footrace, were used instead of the expiration of a candle. This type of auction was first mentioned in 1641 in the records of the House of Lords. The practice rapidly became popular, and in 1660 Samuel Pepys' diary recorded two occasions when the Admiralty sold surplus ships "by an inch of candle". Pepys also relates a hint from a highly successful bidder who had observed that, just before expiring, a candle-wick always flares up sligh…

Read more on Wikipedia

By increasing visibility of an item and therefore demand, auctions can make an extremely rare item more likely to sell for a higher price.

In 2008, the US National Auctioneers Association reported that the gross revenue of the auction industry for that year was approximately $268.4 billion, with the fastest growing sectors being agricultural, machinery, equipment, and residential real estate auctions.

The auctions with the largest revenue for the government are often spectrum auctions (typical revenue is estimated in billions of euros) and quota auctions. In 2019, Russia's crab quota was auctioned for €2 billion. Between 1999 and 2002, the British government auctioned off their gold reserves, raising approximately $3.5 billion.

The most expensive item to ever be sold in an auction is Leonardo da Vinci's Salvator Mundi in 2017 ($450.3 million).

In 2018, the yearly revenues of the two biggest auction houses were $5 billion (Christie's) and $4 billion (Sotheby's).

Auctions come in a variety of types and categories, which are sometimes not mutually exclusive. Typification of auctions is considered to be a part of Auction theory. The economists Paul Milgrom and Robert B. Wilson were awarded the 2020 Nobel Prize for the introduction of new auction types (or formats). Auction types share features, which can be summarized into the following list.
Auctions can differ in the number and type of participants. There are two types of participants: a buyer and a seller. A buyer pays to acquire a certain good or service, while a seller offers goods or services for money or barter exchange. There can be single or multiple buyers and single or multiple sellers in an auction. If just one seller and one buyer are participating, the process is not considered to be an auction.

The forward auction is the most common type of auction — a seller offers item(s) for sale and expects the highest price. A reverse auction is a type of auction in which the roles of the buyer and the seller are reversed, with the primary objective to drive purchase prices downward. While ordinary auctions provide suppliers the opportunity to find the best price among interested buyers, reverse auctions and buyer-determined auctions give buyers a chance to find the lowest-price supplier. During a reverse auction, suppliers may submit multiple offers, usually as a response to competing suppliers' offers, bidding down the price of a good or service to the lowest price they are willing to receive. A reverse price auction is not necessarily 'descending-price' — the reverse Dutch auction is an ascending-price auction because forward Dutch auctions are descending. By revealing the competing bids in real-time to every participating supplier, reverse auctions promote "information transparency". This, coupled with the dynamic bidding process, improves the chances of reaching the fair market value of the item.

A double auction is a combination of both forward and reverse auctions. A Walrasian auction or Walrasian tâtonnement is a double auction in which the auctioneer takes bids from both buyers and sellers in a market of multiple goods. The auctioneer progressively either raises or drops the current proposed price depending on the bids of both buyers and sellers, the auction concluding when supply and demand exactly balance. As a high price tends to dampen demand while a low price tends to increase demand, in theory there is a particular price somewhere in the middle where supply and demand will match. A Barter double auction is an auction where every participant has a demand and an offer consisting of multiple attributes and no money is involved. For the mathematical modelling of satisfaction level, Euclidean distance is used, where the offer and demand are treated as vectors.

Read more on Wikipedia