In this paper, we first explain why human-like dialogue understanding is so difficult for artificial intelligence (AI). We discuss various methods for testing the understanding capabilities of dialogue systems. Our review of the development of dialogue systems over five decades focuses on the transition from closed-domain to open-domain systems and their extension to multi-modal, multi-party and multi-lingual dialogues. From being somewhat of a niche topic of AI research for the first 40 years, it has made newspaper headlines in recent years and is now being discussed by political leaders at events such as the World Economic Forum in Davos. We ask whether large language models are super-parrots or a milestone towards human-like dialogue understanding and how they relate to what we know about language processing in the human brain. Using ChatGPT as an example, we present some limitations of this approach to dialogue systems. Finally, we present some lessons learned from our 40 years of research in this field about system architecture principles: symmetric multi-modality, no presentation without representation and anticipation feedback loops. We conclude with a discussion of grand challenges such as satisfying conversational maxims and the European Language Equality Act through massive digital multi-linguality—perhaps enabled by interactive machine learning with human trainers.This article is part of a discussion meeting issue 'Cognitive artificial intelligence'.