This thesis gives an overview of the currently most mature key encapsulation mecha- nisms (KEMs) based on the theory of error correcting codes. It includes an introduc- tion to the theory of error correcting codes in so much as it applies to these systems and how it can be used to encapsulate keys through a public key (PK) cryptosystem. In order to add context to the KEMs, first the required basics of coding theory and a selection of some of the most common error correcting codes are covered. Then, we revisit public key cryptosystems, key encapsulation, and the security threat models that are being used. This is followed by a thorough description of the current NIST candidates for KEM using post-quantum cryptography: Classic McEliece, BIKE, LEDAcrypt, and HQC.