This paper examines the extent to which large language models (LLMs)  have developed higher-order theory of mind (ToM) ; the human ability to reason about multiple mental and emotional states in a recursive manner (e.g. Ithinkthat youbelievethat sheknows) . This paper builds on prior work by introducing a handwritten test suite – Multi-Order Theory of Mind Q&A – and using it to compare the performance of five LLMs to a newly gathered adult human benchmark. We find that GPT-4 and Flan-PaLM reach adult-level and near adult-level performance on ToM tasks overall, and that GPT-4 exceeds adult performance on 6th order inferences. Our results suggest that there is an interplay between model size and finetuning for the realisation of ToM abilities, and that the best-performing LLMs have developed a generalised capacity for ToM. Given the role that higher-order ToM plays in a wide range of cooperative and competitive human behaviours, these findings have significant implications for user-facing LLM applications.