Computational argumentation has become an essential tool in various fields, including artificial intelligence, law, and public policy. It is an emerging research field in natural language processing (NLP) that attracts increasing attention. Research on computational argumentation mainly involves two types of tasks: argument mining and argument generation. As large language models (LLMs) have demonstrated strong abilities in understanding context and generating natural language, it is worthwhile to evaluate the performance of LLMs on various computational argumentation tasks. This work aims to embark on an assessment of LLMs, such as ChatGPT, Flan models and LLaMA2 models, under zero-shot and few-shot settings within the realm of computational argumentation. We organize existing tasks into 6 main classes and standardise the format of 14 open-sourced datasets. In addition, we present a new benchmark dataset on counter speech generation, that aims to holistically evaluate the end-to-end performance of LLMs on argument mining and argument generation. Extensive experiments show that LLMs exhibit commendable performance across most of these datasets, demonstrating their capabilities in the field of argumentation. We also highlight the limitations in evaluating computational argumentation and provide suggestions for future research directions in this field.