The rapid advancement in quantum computing has led toward a probabilistic threat to conventional cryptographic systems. It has led to innovations in lattice-based cryptography (LBC) as a strong candidate for post-quantum cryptography (PQC). LBC relies on mathematical complexity of lattice problems to ensure security against quantum computers. This paper focuses on recent achievements and fundamentals in lattice-based cryptography including implementations like NTRU and Kyber scheme (lattice-based key encapsulation mechanism that has gained significant attention in recent years), algebra, and hard problems in lattice-based cryptography like Learning with error problems, and NIST standardization process for the future of secure communications in a post-quantum world.