Antimicrobial peptides (AMPs) are promising candidates to combat multidrug-resistant pathogens. However, the high cost of extensive wet-lab screening has made AI methods for identifying and designing AMPs increasingly important, with machine learning (ML) techniques playing a crucial role. AI approaches have recently revolutionised this field by accelerating the discovery of new peptides with anti-infective activity, particularly in preclinical mouse models. Initially, classical ML approaches dominated the field, but recently there has been a shift towards deep learning (DL) models. Despite significant contributions, existing reviews have not thoroughly explored the potential of large language models (LLMs), graph neural networks (GNNs) and structure-guided AMP discovery and design. This review aims to fill that gap by providing a comprehensive overview of the latest advancements, challenges and opportunities in using AI methods, with a particular emphasis on LLMs, GNNs and structure-guided design. We discuss the limitations of current approaches and highlight the most relevant topics to address in the coming years for AMP discovery and design.