We present ``Ouroboros Praos'', a proof-of-stake blockchain protocol that, for the first time, provides security against fully-adaptive corruption in the semi-synchronous setting: Specifically, the adversary can corrupt any participant of a dynamically evolving population of stakeholders at any moment as long the stakeholder distribution maintains an honest majority of stake; furthermore, the protocol tolerates an adversarially-controlled message delivery delay unknown to protocol participants. To achieve these guarantees we formalize and realize in the universal composition setting a suitable form of forward secure digital signatures and a new type of verifiable random function that maintains unpredictability under malicious key generation. Our security proof develops a general combinatorial framework for the analysis of semi-synchronous blockchains that may be of independent interest. We prove our protocol secure under standard cryptographic assumptions in the random oracle model.