Visual object counting has recently shifted towards class-agnostic counting (CAC), which addresses the challenge of counting objects across arbitrary categories—a crucial capability for flexible and generalizable counting systems. Unlike humans, who effortlessly identify and count objects from diverse categories without prior knowledge, most existing counting methods are restricted to enumerating instances of known classes, requiring extensive labeled datasets for training and struggling in open-vocabulary settings. In contrast, CAC aims to count objects belonging to classes never seen during training, operating in a few-shot setting. In this paper, we present the first comprehensive review of CAC methodologies. We propose a taxonomy to categorize CAC approaches into three paradigms based on how target object classes can be specified: reference-based, reference-less, and open-world text-guided. Reference-based approaches achieve state-of-the-art performance by relying on exemplar-guided mechanisms. Reference-less methods eliminate exemplar dependency by leveraging inherent image patterns. Finally, open-world text-guided methods use vision-language models, enabling object class descriptions via textual prompts, offering a flexible and promising solution. Based on this taxonomy, we provide an overview of 30 CAC architectures and report their performance on gold-standard benchmarks, discussing key strengths and limitations. Specifically, we present results on the FSC-147 dataset, setting a leaderboard using gold-standard metrics, and on the CARPK dataset to assess generalization capabilities. Finally, we offer a critical discussion of persistent challenges, such as annotation dependency and generalization, alongside future directions. We believe this survey offers a valuable resource for researchers, capturing the evolution of CAC and providing insights to guide future developments in the field. • Provides the first comprehensive survey on class-agnostic object counting (CAC). • Proposes a taxonomy: reference-based, reference-less, and open-world text-guided. • Benchmarks 30 CAC methods on FSC-147 and CARPK, setting a leaderboard. • Discusses trade-offs between accuracy, flexibility, and annotation requirements. • Identifies open challenges and future directions for generalizable counting.