• We reviewed recent advances in epitaxial growth of elemental two-dimensional materials on coinage metals. • Surface alloys play key roles in both the layer-by-layer and segregation growth of these novel 2D materials. • The periodic table and phase diagram offer us valuable information for selecting suitable substrates. • Surface alloys with stable structures can be used for the heterogeneous growth of either monoelemental or binary 2D materials. Recent achievements in the fabrication of two-dimensional (2D) materials by molecular beam epitaxy have demonstrated that surface alloys are critical intermediates for either layer by layer or segregation growth. The resulting films show a homogenous growth mode that strongly depends on the lattice and properties of surface alloys. Learning from the periodic table, we address the importance of surface alloys in the heterogeneous growth of 2D materials. We demonstrate basic rules in the formation of surface alloys that are correlated with their periodic properties. We highlight the possibility of using 2D topological insulators as the substrates to accomplish quasifreestanding monolayers. Finally, we assess the current status and challenges of binary 2D materials and discuss prospects of their fabrication via surface alloy processes.