Counting the number of items in a visual scene remains a fundamental yet challenging task in computer vision. Traditional approaches to solving this problem rely on domain-specific counting architectures, which are trained using datasets annotated with a predefined set of object categories. However, recent progress in creating large-scale multimodal vision-language models (VLMs) suggests that these domain-general architectures may offer a flexible alternative for open-set object counting. In this study, we therefore systematically compare the performance of state-of-the-art specialized counting architectures against VLMs on two popular counting datasets, as well as on a novel benchmark specifically created to have a finer-grained control over the visual properties of test images. Our findings show that most VLMs can approximately enumerate the number of items in a visual scene, matching or even surpassing the performance of specialized computer vision architectures. Notably, enumeration accuracy significantly improves when VLMs are prompted to generate intermediate representations (i.e., locations and verbal labels) of each object to be counted. Nevertheless, none of the models can reliably count the number of objects in complex visual scenes, showing that further research is still needed to create AI systems that can reliably deploy counting procedures in realistic environments.