The objectives of this thesis is to explore analytical solutions of velocity field, shear stress and temperature distribution subject to the electrically conducting flows of fractionalized non-Newtonian fluids embedded with porous medium. The mathematical modeling of governing equations for fluid flow has been established in terms of fractional derivatives and solved by employing discrete Laplace, Fourier Sine and Hankel transforms. The newly defined fractional derivatives namely Atangana-Baleanu and Caputo fractional derivatives have been implemented on the problems of fluid flows. The general solutions have been investigated under the influence of fractional and non-fractional (ordinary) parameters, magnetohydrodynamics (MHD), porous medium, heat and mass transfer and nanoparticles suspended in base fluids. The obtained solutions satisfy initial, boundary and natural conditions, expressed in terms of special functions and have been reduced for special and limiting cases as well. Moreover, influence of magnetic field, porosity, fractional parameter, heat and mass transfer, nanoparticles and different rheological parameters of practical interest have been investigated. At the end, in order to highlight the differences and similarities among various rheological parameters, the graphical illustration has been depicted for fluid flows.