Cryptanalysis, the study of analyzing and breaking cryptographic algorithms, has long been a crucial aspect of securing digital communications. As quantum computing approaches practical implementation, the cryptographic landscape is poised for significant transformation. This paper delves into the cryptanalysis of both pre-quantum and post-quantum cryptographic systems, highlighting the vulnerabilities and strengths of various algorithms in the face of evolving threats. Pre-quantum cryptography, which includes widely used systems like RSA and Elliptic Curve Cryptography (ECC), is based on mathematical problems that are hard to solve using classical computers. However, quantum algorithms, such as Shor's algorithm, pose a direct threat to these systems by potentially breaking their security within a feasible timeframe. In response, post-quantum cryptography is being developed to offer resistance against quantum attacks. This paper examines the cryptanalysis techniques applicable to these new algorithms, including lattice-based, hash-based, code-based, and multivariate polynomial cryptosystems.