Nonlinear optics has long been studied as a basis for realizing fast information processing devices and sensors. Nonlinear photonic logic enables the elimination of slow electron-photon transduction processes and supports high signal bandwidth at optical carrier frequencies. Recent breakthroughs in material science have ushered in a new era of research on atomically thin 2D materials with strong and novel properties for nonlinear optics. 2D materials may be incorporated with scalable nanophotonic technologies via post-lithographic integration. This thesis presents fundamentals for nonlinear optical properties of the 2D materials and their application for nonlinear quantum optics. It first describes the electronic band structures of two prominent 2D materials, namely, graphene and monolayer MoS2. Their linear and nonlinear optical properties are analyzed and presented. The analysis of graphene's optical property uses an innovative first-order perturbative S-matrix formalism that systematically identifies various physical mechanisms contributing towards the optical Kerr nonlinearity. On the other hand, the analysis of the optical property of monolayer MoS2 adopts a massive Dirac Hamiltonian that leads to linear and nonlinear optical susceptibilities through a standard perturbative calculation. It turns out that, although its real Kerr nonlinear susceptibility is enormously large compared to bulk materials, graphene has an even larger imaginary Kerr nonlinear susceptibility that degrades coherence via strong two-photon absorption. Hence, graphene is not a suitable material for applications where coherence is essential. In contrast, monolayer MoS2 that has non-zero finite bandgap energy turns out to be a simultaneously suitably coherent and highly nonlinear material as its real-to-imaginary ratio of the Kerr nonlinear susceptibility can be adjusted through detuning the optical carrier frequency. The thesis also presents a metamaterial configuration based on Kerr nonlinearity of the monolayer MoS2 coupled to a local surface plasmon. The unique combination of a strong field enhancement from the plasmonic effect and the atomic thickness of highly nonlinear 2D materials constitutes an optical nonlinear oscillator. This system produces highly quantum behavior, namely, photon antibunching and non-Gaussianity. When built on rapidly developing nanophotonic platforms, 2D materials are promising nonlinear optical materials that have a vast potential for a future large-scale quantum information processing network.