The production of quantum computers with fast processing power and practical use will adversely affect the foundations of secure communication, especially for public-key cryptosystems (PKCs). Potential problems and solution scenarios are created early to design quantum-resistant PKCs. In this paper, the emergence of the concept of post-quantum cryptography (PQC) and early precautionary actions are explained. In addition, cryptosystem families that are known/believed to be secure in the presence of quantum computers are discussed. Based on these classes, the selected encryption/key encapsulation (KEM) and signature schemes of the NIST PQC process are expressed. Finally, some open problems for the post-quantum era are summarized.