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Epitaxial growth, a crystallographically oriented growth induced by the chemical bonding between crystalline substrate and atomic building blocks, has been a key technique in the thin-film and heterostructure applications of semiconductors. However, the epitaxial growth technique is limited by different lattice mismatch and thermal expansion coefficients of dissimilar crystals. Two-dimensional (2D) materials with dangling bond-free van der Waals surfaces have been used as growth templates for the hetero-integration of highly mismatched materials. Moreover, the ultrathin nature of 2D materials also allows for remote epitaxial growth and confinement growth of quasi-2D materials via intercalation. Here, we review the hetero-dimensional growth on 2D substrates: van der Waals epitaxy (vdWE), quasi vdWE, and intercalation growth. We discuss the growth mechanism and fundamental challenges for vdWE on 2D substrates. We also examine emerging vdWE techniques that use epitaxial liftoff and confinement epitaxial growth in detail. Finally, we give a brief review of radiation effects in 2D materials and contrast the damage induced with their 3D counterparts.