Large Language Models (LLMs) have transformed Natural Language Processing (NLP) and Software Engineering by fostering innovation, streamlining processes, and enabling data-driven decision-making. Recently, the adoption of LLMs in time-series analysis has catalyzed the emergence of time-series LLMs, a rapidly evolving research area. Existing reviews provide foundational insights into time-series LLMs but lack a comprehensive examination of recent advancements and do not adequately address critical challenges in this domain. This Systematic Literature Review (SLR) bridges these gaps by analysing state-of-the-art contributions in time-series LLMs, focusing on architectural innovations, tokenisation strategies, tasks, datasets, evaluation metrics, and unresolved challenges. Using a rigorous methodology based on PRISMA guidelines, over 700 studies from 2020 to 2024 were reviewed, with 59 relevant studies selected from journals, conferences, and workshops. Key findings reveal advancements in architectures and novel tokenization strategies tailored for temporal data. Forecasting dominates the identified tasks with 79.66% of the selected studies, while classification and anomaly detection remain underexplored. Furthermore, the analysis reveals a strong reliance on datasets from the energy and transportation domains, highlighting the need for more diverse datasets. Despite these advancements, significant challenges persist, including tokenization inefficiencies, prediction hallucinations, and difficulties in modelling long-term dependencies. These issues hinder the robustness, scalability, and adaptability of time-series LLMs across diverse applications. To address these challenges, this SLR outlines a research roadmap emphasizing the improvement of tokenization methods, the development of mechanisms for capturing long-term dependencies, the mitigation of hallucination effects, and the design of scalable, interpretable models for diverse time-series tasks.