The growing interest in eXplainable Artificial Intelligence (XAI) has stimulated research on models with built-in interpretability, among which part-prototype models are particularly prominent. Part-Prototype Models (PPMs) classify inputs by comparing them to learned prototypes and provide human-understandable explanations of the form "this looks like that". Despite this intrinsic interpretability, PPMs have not yet emerged as a competitive alternative to post-hoc explanation methods. This survey reviews work published between 2019 and 2025 and derives a taxonomy of the challenges faced by current PPMs. The analysis reveals a diverse set of open problems. The main issue concerns the quality and number of learned prototypes. Further challenges include limited generalization across tasks and contexts, as well as methodological shortcomings such as non-standardized evaluation. Five broad research directions are identified: improving predictive performance, developing theoretically grounded architectures, establishing frameworks for human-AI collaboration, aligning models with human concepts, and defining robust metrics and benchmarks for evaluation. The survey aims to stimulate further research and promote intrinsically interpretable models for practical applications. A curated list of the surveyed papers is available at https://github.com/aix-group/ppm-survey.