Object counting is an important computer vision application and research topic, which typically involves enumerating the number of objects in an image. Methodologies spanning a broad set of strategies have been proposed for solving object counting problems. These methods have seen an increase in relevance with the recent emergence of several highly successful deep learning techniques, which have led to significant performance improvements on a growing number of annotated counting benchmark datasets. However, despite the recent advancements in deep learning and computer vision, object counting remains a challenging problem with several open research directions. Datasets often contain objects that are highly occluded and which occur across a range of scales and perspectives. Further, popular annotation strategies, like density map annotations, suffer from annotator noise and inconsistency, which creates a performance bottleneck. These annotation strategies also have a high annotation burden, which leads to datasets that are very small when compared to common benchmark datasets in domains like image classification. Given both the significant progress and continued challenges of object counting, this task continues to be an interesting and ongoing research problem. This overview explores the historical context of object counting methods, the fundamental methodologies driving progress, the state of the art methods, and the significant open problems. In particular, we focus on recent trends that attempt to alleviate the problem of the annotation burden for object counting problems.