Large Language Models have shown tremendous performance on a large variety of natural language processing tasks, ranging from text comprehension to common sense reasoning. However, the mechanisms responsible for this success remain unknown, and it is unclear whether LLMs can achieve human-like cognitive capabilities or whether these models are still fundamentally limited. Abstract reasoning is a fundamental task for cognition, consisting of finding and applying a general pattern from few data. Evaluating deep neural architectures on this task could give insight into their potential limitations regarding reasoning and their broad generalisation abilities, yet this is currently an under-explored area. In this paper, we perform extensive evaluations of state-of-the-art LLMs on abstract reasoning tasks, showing that they achieve very limited performance in contrast with other natural language tasks, and we investigate the reasons for this difference. We apply techniques that have been shown to improve performance on other NLP tasks and show that in most cases their impact on abstract reasoning performance is limited. In the course of this work, we have generated a new benchmark for evaluating language models on abstract reasoning tasks.