This dissertation presents theoretical investigations of two novel functional materials, TlBr and MoS2. TlBr is interesting due to its potential application as a high-quality room temperature radiation detector. The stability and electronic structures of three phases of TlBr are studied using Density Functional Theorem (DFT) calculations employing a hybrid functional. The calculated band gaps from DFT with the hybrid functional are in excellent agreement with experimental measurements. The influence of some interstitial and substitutional dopants on TlBr properties is further studied. DFT predicts that interstitial and substitutional C, N, and O dopants in TlBr can display large, localized magnetic moments. A simple model that employs Pauli exclusion principle and group theory is introduced to explain the magnitude of the moments. Van der Waals (VDW) bonded, layered transition metal dichalcogenides (TMDCs) are promising 2D materials and have drawn much research attention recently. A theory for the epitaxial growth on the transition metal dichalcogenides is developed and analyzed. A prototypical example of Au epitaxially grown on MoS2 is studied. It is demonstrated that if one accounts for interfacial energies and strains, the presence of misfit dislocations, and the compliance of the MoS2 substrate, the experimentally observed growth orientation (Au {111} orientation) will be energetically favored. Meanwhile, with only the top substrate layer being compliant (strained), this epitaxy method can serve as a means to exfoliate and transfer large single layers sheets of MoS2. The exfoliation of monolayer based on this epitaxy idea has been realized experimentally. This epitaxy model could prove technologically useful in 2D electronic devices and in epitaxial thin film growth applications.