The rapid evolution of communication networks in recent decades has intensified the need for advanced Network and Service Management (NSM) strategies to address the growing demands for efficiency, scalability, enhanced performance, and reliability of these networks. Large Language Models (LLMs) have received tremendous attention due to their unparalleled capabilities in various Natural Language Processing (NLP) tasks and generating context-aware insights, offering transformative potential for automating diverse communication NSM tasks. Contrasting existing surveys that consider a single network domain, this survey investigates the integration of LLMs across different communication network domains, including mobile networks and related technologies, vehicular networks, cloud-based networks, and fog/edge-based networks. First, the survey provides foundational knowledge of LLMs, explicitly detailing the generic transformer architecture, general-purpose and domain-specific LLMs, LLM model pre-training and fine-tuning, and their relation to communication NSM. Under a novel taxonomy of network monitoring and reporting, AI-powered network planning, network deployment and distribution, and continuous network support, we extensively categorize LLM applications for NSM tasks in each of the different network domains, exploring existing literature and their contributions thus far. Then, we identify existing challenges and open issues, as well as future research directions for LLM-driven communication NSM, emphasizing the need for scalable, adaptable, and resource-efficient solutions that align with the dynamic landscape of communication networks. We envision that this survey serves as a holistic roadmap, providing critical insights for leveraging LLMs to enhance NSM.