Quantifying the number of individuals in images or videos to estimate crowd density is a challenging yet crucial task with significant implications for fields such as urban planning and public safety. Crowd counting has attracted considerable attention in the field of computer vision, leading to the development of numerous advanced models and methodologies. These approaches vary in terms of supervision techniques, network architectures, and model complexity. Currently, most crowd counting methods rely on fully supervised learning, which has proven to be effective. However, this approach presents challenges in real-world scenarios, where labeled data and ground-truth annotations are often scarce. As a result, there is an increasing need to explore unsupervised and semi-supervised methods to effectively address crowd counting tasks in practical applications. This paper offers a comprehensive review of crowd counting models, with a particular focus on semi-supervised and unsupervised approaches based on their supervision paradigms. We summarize and critically analyze the key methods in these two categories, highlighting their strengths and limitations. Furthermore, we provide a comparative analysis of prominent crowd counting methods using widely adopted benchmark datasets. We believe that this survey will offer valuable insights and guide future advancements in crowd counting technology.