Recent results from theoretical and computational materials science research will be reported on the topics of surface engineering by action of externally applied fields and optimal design of two-dimensional (2D) graphene-based nanomaterials. On the first topic, we address problems of surface morphological instabilities in stressed elastic solid material systems used in electronic and nanofabrication technologies and develop surface stabilization strategies by exploiting the action of externally applied fields, such as electric fields and thermal gradients. We also explore the patterns that emerge as a result of such stress-induced instabilities toward fine-length-scale surface engineering. In addition, we introduce the current-driven dynamics of single-layer islands on conducting substrate surfaces as a directed assembly strategy for surface nanopatterning. These studies are based on atomistically informed continuum-scale models, as well as linear and nonlinear stability theories and bifurcation theories based on these models, in conjunction with self-consistent dynamical simulations involving boundary-integral methods for solution of boundary-value problems coupled with front tracking methods for propagation of domain boundaries. On the second topic, we establish structure-property relations in graphene-based nanomaterials, which can be used for precise tuning of electronic, mechanical, and thermal properties of such materials. We focus on superstructures of diamond nanocrystals (SDNs) embedded between the graphene planes of twisted bilayer graphene, synthesized by patterned chemical functionalization of graphene bilayers, and graphene nanomeshes (GNMs), which are regular nanoporous 2D material structures. We demonstrate the tunability of the electronic band gap, the mechanical properties, and the fracture behavior of SDNs achieved by precise control of the extent and pattern of chemical functionalization. We also demonstrate the tunability of the thermal conductivity of GNMs by controlling the GNM porosity, as well as pore and pore arrangement parameters. Our 2D nanomaterials studies rely on implementations of first- principles density functional theory for calculations of atomic structure and electronic band structure, molecular-dynamics (MD) simulations of dynamic deformation tests, and non-equilibrium MD simulations of thermal transport.