A decentralized exchange (DEX) is a cryptocurrency exchange which operates in a decentralized way, i.e., without a central authority. Decentralized exchanges allow peer-to-peer trading of cryptocurrencies.
Because users do not need to transfer their assets to the exchange, decentralized exchanges reduce the risk of theft from hacking of exchanges. Decentralized exchanges can also prevent price manipulation or faked trading volume through wash trading, and are more anonymous than exchanges which implement know your customer requirements.
There are some signs that decentralized exchanges have been suffering from low trading volumes and market liquidity. The 0x project, a protocol for building decentralized exchanges with interchangeable liquidity attempts to solve this issue. Totle is another project trying to remedy the lack of liquidity by pooling together leading decentralized exchanges and routing users' trades according to best price. 
Degrees of Decentralization
A decentralized exchange can still have centralized components, whereby some control of the exchange is still in the hands of a central authority. A notable example being IDEX blocking New York State users from placing orders on the platform.
In July 2018, decentralized exchange Bancor was reportedly hacked and suffered a loss of $13.5M USD in assets before freezing funds. In a Tweet, Charlie Lee, the creator of Litecoin spoke out and claimed an exchange cannot be decentralized if it can lose or freeze customer funds.
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