To solve the threat of quantum computing to the security of public-key cryptography, post-quantum cryptography has become a frontier focus in the field of cryptography. Post-quantum cryptography guarantees the security of the algorithm through mathematical theories, but it is vulnerable to side-channel attacks in specific implementation and applications, which will seriously threaten the security of post-quantum cryptography. This study is based on the round 2 candidates in the NIST post-quantum cryptography standardization process and the round 2 candidates in the CACR public key cryptography competition in China. First, classification investigations of various post-quantum cryptographic algorithms are conducted, including lattice-based, code-based, hash-based, and multivariate-based cryptographic algorithms. Then, their security status against side-channel attacks and existing protection strategies are analyzed. To analyze the methods of side-channel attack against post-quantum cryptography, it is summarized that the commonly used post-quantum cryptography side-channel attack methods, attack targets, and attack evaluation indexes for various post-quantum cryptography according to the classification of core operators and attack types. Furthermore, following the attack types and attack targets, the existing countermeasures for attack and the costs of defense strategies are sorted out. Finally, in the conclusion part, some security suggestions are put forward according to the attack method, protection means, and protection cost, and also the potential side-channel attack methods and defense strategies in the future are analyzed.