The rapid maturation of quantum computing architectures has elevated the security of classical cryptographic infrastructure from a theoretical concern to an immediate strategic risk. This research review provides a comprehensive scholarly analysis of quantum cryptography and post-quantum security mechanisms, grounded exclusively in literature, standards, and experimental findings available during 2019 and 2020. The paper begins by tracing the evolution of classical cryptography and its structural vulnerabilities in the face of quantum computation, with detailed technical treatment of Shor's algorithm — which renders RSA, Diffie-Hellman, and elliptic-curve cryptography computationally insecure — and Grover's algorithm, which provides a quadratic speedup for symmetric-key exhaustion. The review then surveys the field of Quantum Key Distribution (QKD), examining the BB84 and E91 protocols with analytical depth, including their security proofs, practical implementations, and deployment constraints. The post-quantum cryptography (PQC) landscape is systematically assessed across four major algorithmic families — lattice-based, hash-based, code-based, and multivariate cryptography — evaluated against the backdrop of the NIST PQC standardization process as it stood at the close of Round 2 in early 2020. Hardware platforms relevant to quantum computing in this period, including IBM Quantum processors, Google's Sycamore quantum supremacy demonstration of 2019, and D-Wave's annealing systems, are discussed in terms of their implications for cryptographic threat timelines. Comparative analysis is presented through four structured tables. The paper concludes with a forward-looking research agenda grounded in the understanding of 2019–2020, identifying migration strategy, hybrid cryptographic architectures, and QKD network scaling as priority challenges.