This chapter establishes the theoretical foundations and state of the art in blockchain consensus mechanisms. It begins with core system models, distinguishing between permissionless and permissioned architectures, and examines network models crucial for consensus protocol design. The chapter analyzes fundamental requirements for blockchain consensus, including security, scalability, decentralization, and adaptability, while introducing essential cryptographic concepts such as threshold signatures and balanced graph edge partitioning. A comprehensive review of existing blockchain systems follows, examining three main categories: PoW-based approaches that explore decoupling functionalities, parallel chains, and DAG structures; PoS-based systems focusing on chain-based and BFT-based protocols; and sharding-based solutions addressing scalability through parallel processing. This extensive review of related work provides the theoretical framework necessary for understanding the adaptive consensus protocols presented in subsequent chapters.