The security of long-standing cryptographic protocols such as the RSA algorithm is under threat from the emergence of quantum computing, which is transforming the landscape of cryptography. The interaction between RSA and quantum-based threats—specifically, Shor’s Algorithm—is examined in this review. The paper emphasizes how RSA’s security depends on the challenge of factoring big integers after providing a thorough analysis of the algorithm’s creation, encryption, and decryption steps. Large integer factoring is made possible by quantum computing, which breaks this security paradigm by using concepts like superposition. This process is accelerated by Shor’s Algorithm, a quantum instrument that poses a danger to RSA encryption. To counter this threat, the paper emphasizes how urgent it is and supports post-quantum cryptography alternatives. The influence of Shor’s Algorithm on RSA at different key lengths is examined along with vulnerability assessments.