Curve Finance is a decentralized exchange created by Michael Egorov. It is currently the largest protocol by total value locked. The original white paper for Curve Finance was written on November 10th, 2019 and published on January 3rd, 2020. Curve Finance is a DAO governed by the CRV token which launched in August 2020.
The Curve Finance white paper, originally named StableSwap was written by Russian physicist, Michael Egorov on November 10th, 2019. The protocol launched a few weeks later on January 3rd, 2020 and grew organically to around $500,000 TVL within 10 days. At the time, Uniswap and Kyber were major players in the decentralized exchange space but neither protocols had native support for pegged assets which usually made it more expensive to trade such assets.
The original pool had to be relaunched after a major vulnerability disclosure by Sam Sun. Both original pools used yield-bearing tokens from Compound Finance after the suggestion was made by Compound Finance founder Robert Leshner. Curve smart contracts were subsequently audited by Trail of Bits and Sam Sun.
iEarn and yTokens
Curve Finance third pool included ytokens, yield bearing tokens developed by Andre Cronje in early 2020 which later spawned Yearn Finance. Those yTokens would rebalance themselves to the lending protocol with the better yield (usually AAVE or Compound).
In March 2020, a hacking attempt on the BUSD pool left it imbalanced causing panic among Curve Finance early liquidity providers. Andre Cronje was held responsible despite not profiting or causing the imbalance. He later announced he was leaving the DeFi community on the 3rd of March 2020.
In April, another bug related to a logic issue with the underlying rate of ytokens was discovered in a newly launched sUSD pool which lead to a shut down. No funds were lost but Curve Finance was seen as a riskier protocol which lead to a rework of security practices. In a bug disclosure, Michael Egorov said: "Never deploy a pool with unaudited changes: no matter how small and obvious they are;" 
Native Bitcoin and Yield Farming
By early June, Curve Finance TVL had grown to over $25m in TVL with daily volumes as high as $47m. The team had grown to five people with the addition of Ben Hauser, a Vyper contributor who developed a testing framework for Ethereum called Brownie, Charlie Watkins who became the second employee, anonymous contributor Kendrick Llama and Taiwan based content creator Chris.
The protocol also began supporting native Bitcoin swaps and deposits with the help of Ren Protocol with both protocols being widely credited for the advancement of Bitcoin on Ethereum.
It was at the same time Kain Warwick of the Synthetix protocol suggested incentivizing the sUSD pool with the SNX token, a move that marked the creation of yield farming.
In June, it was rumoured Curve Finance would transition to DAO with the help of the CRV token, a governance and utility token. The token launch happened on the 13th of August after an anonymous developer going by the name of the 0xc4ad took it upon himself to develop the finalized contracts which had been finalized and made public days earlier on GitHub. A few hours later, Charlie Watkins announced the team would adopt 0xc4ad deployment.
CRV was immediately listed on Binance and other major exchanges. Around a week after the CRV launch, Curve Finance reached $1 billion in TVL and had long shedded its image of a risky protocol.
In late September, Charlie Watkins introduced a proposal to start fee sharing between liquidity providers and token holders which was subsequently accepted by the Curve DAO.
Despite the controversial CRV launch, Curve grew rapidly in the following months and began being referred to as the backbone of DeFi yields as many protocols in the industry rely on the protocol.
It reached $10b TVL for the first time in June 2021.
Convex and bribing
In early February, Convex, a yield optimizer built atop and supported by Curve Finance launched and began growing rapidly accounting for around 90% of Curve Finance TVL on Ethereum. The introduction of a bribing mechanism and the so-called "Curve Wars" allowing protocols to buy voting power from Convex and Curve Finance users to receive emissions from both protocols is largely credited as one of the reasons behind the protocol growth.
Crypto Pools and multi-chain expansion
In 2021, as Ethereum grew more expensive to use, Curve Finance focused on leaner pool implementation as its original lending pools became too costly to use. It launched on its first sidechain in February 2021 as part of its long standing relation with Andre Cronje. It later launched on Polygon in April 2021.
In June 2021, as a major departure from the pegged assets space, the protocol launched a pool with volatile assets called tricrypto. In November 2021, more crypto pools appeared on Curve.
Curve Finance enables low-slippage swaps between stable assets using the algorithm described in its StableSwap whitepaper. Curve's major use cases revolve around maintaining peg for stablecoins and wrapped assets (e.g., wrapped Bitcoin).
Curve operates using pools, which group assets whose prices should be stable, and metapools, which group individual assets paired with an existing pool. Exchanges from one asset to another are routed through these pools. For example, the largest pool for stablecoins is 3pool, which holds DAI, USDC and USDT. To enable and incentivize liquidity for their asset, individual stablecoin issuers have created metapools that group their stablecoin (e.g., FRAX) with the 3pool. This allows stablecoins of varying liquidity access to the liquidity strength of three large and reputable assets.
Curve v2 pools enable support for volatile assets using a different algorithm, described in its crypto pools whitepaper. The major volatile asset pool is tricrypto2, which holds USDT, wBTC and wETH. However, since Curve has allowed creation of crypto pools for any asset, protocols have mostly created pools that pair their assets with ETH or USDC rather than following the stablecoin pattern of pairing with a metapool like tricrypto2.