This work presents a review of the current progress, challenges, and potential future development opportunities for two-dimensional (2D) material-based heterostructures, including their fabrication techniques as well as their applications in various functional devices. The fabrication techniques for 2D material-based vertical heterostructures are first reviewed and discussed, including artificial stacking, chemical vapor deposition (CVD), molecular beam epitaxy, and others. Then, 2D material-based lateral heterostructure growth techniques are reviewed, including CVD and others. Subsequently, various functional device applications based on 2D material-based heterostructures are systematically reviewed and discussed, including electronic devices, optoelectronic devices, electrochemical devices, and others. The advantages and disadvantages for each fabrication/growth technique are compared and analyzed, including those for both vertical and lateral heterostructures. In addition, the current primary challenges for further development of 2D material-based heterostructures and their functional devices are discussed and analyzed, including lack of precise control, low interface/surface quality, surface passivation issue, low light absorption, lack of system integration and application, complex device fabrication process, as well as low device fabrication efficiency. Various potential solutions are proposed that have the potential to overcome these existing primary challenges to achieve better device performance and thus their ultimate industry applications for2D material-based heterostructures.