There has been growing interest during the last decade in two-dimensional (2D) materials due to their important roles in various scientific and technological applications such as detectors, lasers and light emitting diodes. In this thesis we present a theoretical investigation of a couple of such 2D materials from group V monolayers and their compounds. Firstly, ordered alloys of GaxAl1−xN hexagonal monolayer are studied and the effect of Al content on mechanical, electronic, thermal and optical properties are investigated. The optimized lattice constants and band gaps change in accordance to Vegard’s Law. Low barrier energies and favorable substitution of Ga by Al may show feasibility of fabrication. Segregation is also checked with mixing energy calculations. The dynamical stability of alloys is shown by phonon spectrum analysis and MD simulations. GaxAl1−xN alloys give lower in-plane stiffness than h-BN or graphene, but higher Poisson’s ratio than most realized 2D systems. Heat capacity of alloys delivers a decrease with Al content at low temperatures but it converges to the classical limit at high temperatures. The absorption onset of GaxAl1−xN alloys remain in the near UV range and prominent absorption peaks blue-shifts with increasing x in compliance with the variation of the band gap. The considered systems, in regard to their stability and tunable fundamental properties seem to be very promising 2D semiconductors for wide range of applications at reduced scales. Then, the interaction of alkali metal atoms (Li, Na, and K) with single layer and periodic structures of hb-As and sw-As phases are revealed by first-principles methods. Arsenene phases are considered to be used as electrodes (anode) for ion-batteries. Strong alkali-electrode binding and low diffusion energy barriers gives out better cycling stability and faster diffusion, respectively. hb-As shows better storage capacity than sw-As. However, deviations from ordered pattern and decline of formation energy with increasing doping level point out a possible structural transformation. By MD calculations, crystalline to amorphous phase transition is seen even for low concentrations level at ambient temperature. The average open-circuit voltages of 0.68-0.88 V (0.65-0.98 V) with specific capacity up to 715 mAhg−1 (358 mAhg−1) are calculated for single layer (periodic) configurations. Overall, non-crystalline phases are calculated to offer more favorable structures than crystalline configurations and they provide more coherent results when compared with experimental data. The obtained voltage profile together with low diffusion barriers and strong metal-electrode binding suggests arsenene as a promising anode material to be used in for alkali-ion battery applications. Lastly, the formation of dumbbell (DB) geometry upon adsorption of Ga, N adatoms to GaN monolayer is investigated. While Ga-N DBs are unstable, Ga-Ga and N-N DB geometries are predicted to form in an exothermic and spontaneous scheme. Cohesive energy of hexagonal GaN monolayer decreases when a DB is formed on its surface. Electronic structures for Ga-Ga DBs for 2×2, 3×3, 4×4 and 5×5 phases show spinpolarized and degenerate bands mainly contributed by p-orbitals of the atoms in impurity zone. Degenarated bands are not observed for N-N dumbbell for HDP, TDP, 2×2 and 3×3 phases. Upon DB formation, semiconductor GaN monolayer become spin-polarized semiconductor with varying band gap, where this functionalization allows electronic structure to be tuned substantionally. This would be highly desired for nanoscale electronic and optical devices. These Ga-Ga and N-N DB geometries may also be used for the synthesis of layered GaN structures.