Since the first isolation of graphene from graphite by mechanical exfoliation,atomically-thin or layered materials have been occupying the central stage of today's condensed matter physics and materials sciences because of their rich and exotic properties in two dimensions(2D). Many members of the ever-expanding 2D materials family, such as graphene, silicene,phosphorene, borophene, hexagonal boron nitride, transition metal dichalcogenides, and even the strong topological insulators, share the distinct commonality of possessing relatively weak van der Waals (vdW) interlayer coupling, whereas each member may invoke its own fabrication approaches, and is characterized by its unique properties. In this review, we first discuss the major atomistic processes and related morphological evolution in the nonequilibrium epitaxial growth of vdW layered materials. Representative systems covered include the vdW epitaxy of both monolayered 2D systems and their lateral or vertical heterostructures, emphasizing the vital importance of the vdW interactions in these systems. We also briefly highlight some of the recent advances in the property optimization and functionalization of 2D materials, using examples from the fields of optics, electronics, spintronics, and catalysis.