Ouroboros

Proof of Stake algorithm that will be used by Cardano
"Ouroboros is a provably secure proof of stake algorithm developed by a team of IOHK scientists led by Prof. Aggelos Kiayias. The algorithm is backed by a peer reviewed paper. According to Charles Hoskinson, the founder of Cardano, Ouroboros’s paper was rewritten 8 times."
Has three versions, Ouroboros Classic, Ouroboros Praos and Ouroboros Genesis. Is now (9-9-2019) in the Genesis phase.
This post goes deep into how it works without going into mathematics and also on how secure it is.

"Since the ‘update’ of the protocol to the Genesis version, Ouroboros is the first PoS protocol that is mathematically proven to guarantee persistence and liveness in both a synchronous and semi-synchronous setting — under the assumption that a honest majority participating, just like Bitcoin. Hence, it is more secure than other PoS protocols that require at least 2/3 honest participants (e.g. Ethereum Casper, Algorand) and equally secure as Bitcoin, but with a much lower energy expenditure and better performance."

Uses a cryptographic function called Verifiable Random Function (VRF). VRF’s were originally invented by Turing Award winner Silvio Micali, who is currently a professor at MIT and is working on a cryptocurrency called Algorand.