Digital holographic microscopy (DHM) is a rapidly developing imaging technique that allows for the reconstruction of three-dimensional images from holographic data. In this review paper, we present an overview of the state-of-the-art in on-axis DHM reconstruction methods. We classify these methods into three categories: direct, iterative, and machine learning based algorithms. In the direct category, we summarize methods that directly apply the holographic data to reconstruct the sample without any iterative optimization. These methods are fast and efficient, but they often suffer from noise and low contrast. The iterative category contains methods that use an iterative optimization process to improve the quality of the reconstructed image. This can lead to higher accuracy and noise reduction, but at the cost of increased computational complexity. Within the iterative category, we also discuss methods that assume sparsity in the reconstruction and other domains, which can further improve the convergence of the reconstruction algorithm. Finally, we describe machine learning methods for DHM reconstruction. These methods use deep neural networks trained on large datasets to learn the relationship between 2D holographic data and their reconstructed fields. Machine learning based methods can achieve similar results to the iterative methods at a fraction of the time, however, they require large amounts of computational resources and training data. We provide a summary of these methods in a table (Table B.1) and a map (Fig. 1), and unify the naming convention of all methods, allowing readers to easily compare and contrast the different approaches.