Resource-constrained devices play an important role in digital communications. However, their security is difficult because they are highly vulnerable to malicious attacks and require lightweight cryptographic algorithms for security. Recently, the emergence of quantum computers has led to significant setbacks for widely used traditional cryptography schemes. Post-quantum cryptography algorithms represent a development that quantum computers are unable to solve. Post-quantum cryptography (PQC) offers a promising solution, with algorithms designed to withstand quantum attacks. However, implementing PQC in resource-constrained environments presents its own set of challenges. Lattice-based cryptography has emerged as a leading candidate due to its favorable trade-off between security and efficiency. Hash-based methods also offer viable solutions when adapted to smaller key sizes, although challenges remain. Other PQC approaches, such as multivariate, code and isogeny-based schemes, still face issues with key sizes. This paper reviews recent advancements in optimized PQC algorithms tailored for such environments, demonstrating their feasibility and effectiveness. Additionally, it explores future directions for lightweight PQC, providing valuable insights for researchers in the field.