The evolution of 5G and anticipated 6G networks presents new opportunities for connectivity, performance, and services, but it also introduces substantial security risks due to increased complexity and attack surfaces. Traditional perimeter-based security models fall short in these environments, prompting a shift towards Zero Trust Architecture (ZTA), which enforces principles of strict identity verification, least-privilege access, and continuous monitoring. As ZTA gains traction, standardization bodies such as NIST, 3GPP, and ETSI are developing guidelines to incorporate Zero Trust principles into 5G and 6G infrastructures. This paper explores how these standardization efforts, alongside core ZTA principles, provide a resilient and adaptable security framework for 5G and future 6G networks. Furthermore, the paper highlights the challenges of implementing Zero Trust in 5G and 6G networks and suggests future directions for overcoming these obstacles.