The squeezing flow occurs due to the action of normal stresses that originate from different configurations of the plates movement. This context may resound to be profoundly simple from the onset, but has important applications in the areas of engineering, physics, biology, and material sciences. In the past few years, the study of rheometric properties of fluids has garnered profound attention due to its vast industrial applications. Examples include modeling of lubrication systems involved in squeezing of fluids, compression and injection molding processes of metals and polymers, hydrodynamical tools and machines, modeling of chewing and eating, and modeling of the functions of heart valves and blood vessels. As a result, it is a major focus of researchers working in fluid mechanics. This thesis presents the theoretical analysis of unsteady squeezing flow of Newtonian and nonNewtonian fluids between two parallel plates under various boundary conditions. Observation of the squeezing behavior and associated rheological properties of the fluid can be interpreted using various analytical and numerical techniques. Due to the simplicity of the geometry involved, these can also be realized experimentally, while obtaining measurements from sensors or feedback control loops. For the analytical and numerical approach, the squeeze behavior is based on various models like Newtonain, Casson, Power law etc which leads to various differential equations. For this purpose, the thesis goes into a detailed investigation of the rheological properties of fluid flow with various boundary conditions and fluid types. These properties include velocity profiles, pressure distribution, and skin-friction. This investigation starts initially with the analysis of an unsteady flow of Newtonian fluid squeezed between two circular plates with slip and no-slip at the boundaries. It then gradually extends to higher order problems and different boundary conditions. This includes firstly a case involving squeezing of an incompressible Newtonian fluid passing through porous medium, followed by another case involving squeezing of non-Newtonian Casson fluid having magnetohydrodynamical effect, and passing through porous medium. The Casson fluid model is further extended to analyze the slip effect at fluid-solid interface. All these cases are dealt in separate chapters. In all cases, similarity transformations are used for the conversion of PDEs to highly nonlinear ODEs. Various analytical techniques like Optimal Homotopy Asymptotic Method (OHAM), Homotopy Perturbation Method (HPM), Homotopy Perturbation Laplace Method (HPLM), and numerical schemes like Explict and Implicit Runge Kutta Method of order 4 (ERK4 & IRK4) and NDSolve (Mathematica Solver) are applied for the solution and analysis of the modeled problems. Convergence and validity of the obtained analytical solutions are checked by finding various order solutions along with residual errors, and comparing it with numerical results. In addition, the thesis also proposes a novel adaptation to a scheme that combines tradiiii tional perturbation techniques with Homotopy using a Laplace transform. This scheme is applied to a problem of squeezing flow of an incompressible Newtonian fluid through porous medium, and tested against various analytical and numerical schemes.