Consecutive to the isolation of graphene and uncovering its extraordinary properties, the dynasty of two-dimensional (2D) materials has expanded rapidly. A realization of every new member suggests novel features, holding the promise to be used in current and prospective nanodevices. In parallel with the attempts on exploring new 2D systems, the formation of ternary configurations has been suggested as an alternative approach to tailor the inherent properties of the already existing 2D structures. In accordance with recent advancements in ternary 2D systems, in this dissertation, we design and investigate the 2D systems which possess three types of elements in their crystal structure. In this regard, we design Janus Al2XX′ (X/X′: O, S, Se, Te) crystals, 1H, 1T, and 1T′ phases of Janus WXO (X = S, Se, and Te) monolayers and Janus BiXY (X= S, Se, Te, and Y = F, Cl, Br, I ) nanosheets and investigate their structural, vibrational, elastic, piezoelectric, and electronic properties by first-principle methods. In addition, inspired by the synthesis of penta-Si nanoribbons, advancements in 2D pentagonal systems, and recent developments on ternary structures, we propose and investigate a new ternary pentagon-based 2D monolayer, namely penta-BNSi. We study the mechanical, electronic, piezoelectric, photocatalytic, and optical properties of penta-BNSi crystal and reveal its suitability to be used in optoelectronics and photocatalytic applications. Then, we focused our attention on new family of MA2Z4 monolayers. Based on this motivation, we perform a comprehensive study on physical properties of MSi2Z4 (M: Pd and Pt, Z: N and P) monolayers and suggested novel single layer of InSiN2 (In2Si2N4). For this purpose, in both studies, the ground state configurations of the designed materials are determined, and then the dynamical and thermal stability of these nanosheets are investigated using phonon spectrum analysis and ab initio molecular dynamic (AIMD) simulations, respectively. Next, each structure’s Raman and infrared (IR) spectrum are analyzed, and the corresponding atomic displacements of the optical phonon modes are presented. Then, the mechanical properties are studied in terms of in-plane stiffness and Poisson’s ratio. The electronic band structures are computed in the electronic properties section, and the corresponding energy band gaps are reported. For MSi2Z4 (M: Pd and Pt, Z: N and P) monolayers, the optical response is examined via calculation of the complex dielectric function by taking many-body interactions into account. In the final study, the effect of an external biaxial strain on the electronic and vibrational properties of the InSiN2 nanosheet is investigated, and the variation of the obtained properties under strain is illustrated. As a result of a thorough theoretical study focusing on ternary 2D materials, it can be said that the examined crystals are stable systems with potential applications in a wide range of nanoelectronics and nanomechanical devices.