Gallium oxide (Ga2O3) is a promising semiconductor for high power devices and solar blind ultraviolet photodetectors due to its large bandgap, a high breakdown field, and high thermal stability. Recently, a considerable achievement has been obtained for the growth of high-quality β-Ga2O3 and high performance β-Ga2O3 based devices. However, rapid advance in device performance can be limited by the critical issues of metal contacts to β-Ga2O3 such as barrier height, leakage current, ohmic contact, and surface, interfacial and deep states. This article aims to provide a review on the recent studies in the control and understanding of metal contacts to β-Ga2O3, particularly in terms of the barrier formation. This review suggests that understanding the current transport mechanisms of metal contacts to β-Ga2O3 more thoroughly is necessary to enhance the performance, stability and reliability of β-Ga2O3 based devices.