Large Language Models (LLMs) have revolutionized various Natural Language Generation (NLG) tasks, including Argument Summarization (ArgSum), a key subfield of Argument Mining. This paper investigates the integration of state-of-the-art LLMs into ArgSum systems and their evaluation. In particular, we propose a novel prompt-based evaluation scheme, and validate it through a novel human benchmark dataset. Our work makes three main contributions: (i) the integration of LLMs into existing ArgSum systems, (ii) the development of two new LLM-based ArgSum systems, benchmarked against prior methods, and (iii) the introduction of an advanced LLM-based evaluation scheme. We demonstrate that the use of LLMs substantially improves both the generation and evaluation of argument summaries, achieving state-of-the-art results and advancing the field of ArgSum. We also show that among the four LLMs integrated in (i) and (ii), Qwen-3-32B, despite having the fewest parameters, performs best, even surpassing GPT-4o.