Kerr-nonlinearity or third order nonlinearity chi(3) is ubiquitous in nature and is the focus of this dissertation. In the first part of this work, we study one of the outcomes of Kerr-nonlinearity and beam collapse, which we term as “loss of polarization” for intense beams of elliptical input polarization, that undergo self-focusing when traveling through an optical medium with positive Kerr-nonlinearity. We study this phenomenon theoretically and experimentally and show that for such beams, the polarization angle at the output becomes extremely sensitive to arbitrarily small input power fluctuations. We perform experiments in fused silica glass, water and nitrogen gas, and at two different wavelengths (800 nm and 1550 nm) to demonstrate that this phenomenon is universal in nature and more prominent in the anomalous group velocity dispersion (GVD) regime compared to the normal-GVD regime. In the second part, we identify a lack of experimentally measured nonlinear refractive index at mid-infrared (mid-IR) wavelengths for commonly used materials such as sapphire, zinc selenide, etc. and we measure them using the spectrally resolved two-beam coupling (SRTBC) technique developed by Kang, et. al. from the Wise group at Cornell University. We perform measurements of Kerr-nonlinearity for a variety of materials including sapphire (Saph), fused silica (FS), calcium fluoride (CaF2), magnesium fluoride (MgF2), zinc selenide (ZnSe) and zinc sulphide (ZnS) in the mid-IR wavelength range from 2.3-um to 8.3-um. We report negligible dispersion in the nonlinear refractive index n2 measured over this regime. n2 of ZnSe and ZnS is found to be ~ 2 orders of magnitude higher than that of Saph, FS, CaF2 and MgF2, and the measured values of n2 at mid-IR wavelengths are slightly lower than their counterparts in the visible to near-IR regimes in literature. In the last part of our work, we apply the SRTBC technique to measure n2 graphene at 3.5-um. To our knowledge, this is the first time of applying the SRTBC technique to a 2-dimensional (2-D) material and this is the first experiment to measure the graphene nonlinearity in the mid-IR regime beyond 2-um. We electrostatically gate the graphene sample to measure dependence of the Kerr-nonlinearity on the Fermi energy of graphene. Our study shows that the sign and magnitude of graphene nonlinearity is tunable by gating. This work has potential applications in exploring graphene as a phase change material for integrated photonics applications. At the same time, these measurements open doors to future work where the SRTBC technique can be applied to measure nonlinearity of other 2-D materials.