The advent of quantum computers poses a significant threat to the security of classical cryptographic systems. To address this concern, researchers have been actively investigating the development of post-quantum cryptography, which aims to provide encryption schemes that remain secure even in the face of powerful quantum adversaries. To address this serious problem, the National Institute of Standards and Technology (NIST), a body of the US government, has been working on the selection and standardization of cryptographic algorithms through competitive and rigorous evaluation on different fronts. NIST has selected different candidate algorithms to standardize public-key encryption, including key establishment algorithms and digital signature algorithms. This paper reviews some selected cryptosystems, mainly based on lattice- and code-based cryptosystems. These include digital signature algorithms, such as CRYSTALS-Dilithium, code-based cryptosystems, such as McEliece, and key encapsulation methods, specifically, Classic McEliece, BIKE and HQC. We will review these algorithms and discuss their security aspects and the current state-of-the-art in the development of these algorithms post NIST 3rd finalized selection. We will also touch briefly on the differences and practical applications of each of these schema. This review is intended for engineers and practitioners alike.