The inherent susceptibility of low-dimensional materials to thermal fluctuations has long been expected to pose a major challenge to achieve intrinsic long-range ferromagnetic order in two-dimensional materials. The recent explosion of interest in atomically thin materials and their assembly into van der Waals heterostructures has renewed interest in two-dimensional ferromagnetism, which is interesting from a fundamental scientific point of view and also offers a missing ingredient necessary for the realization of spintronic functionality in van der Waals heterostructures. Recently, several atomically thin materials have been shown to be robust ferromagnets. Such ferromagnetism is thought to be enabled by magnetocrystalline anisotropy which suppresses thermal fluctuations. In this article, we review recent progress in two-dimensional ferromagnetism in detail and predict new possible two-dimensional ferromagnetic materials. We also discuss the prospects for applications of atomically thin ferromagnets in novel dissipationless electronics, spintronics, and other conventional magnetic technologies. Particularly, atomically thin ferromagnets are promising to realize time reversal symmetry breaking in two-dimensional topological systems, providing a platform for electronic devices based on the quantum anomalous Hall effect showing dissipationless transport. Our proposed directions will assist the scientific community to explore novel two-dimensional ferromagnetic families which can spawn new technologies and further improve the fundamental understanding of this fascinating area.