Abstarct: Viscous fingering instability is a well-known instability that occurs between the interfaces of two fluids with different viscosities. Creation and propagation of these fingers are playing an important role in displacement of fluids and especially on oil transformation from discovered oil reservoirs. A grab attention effort in this field is allocated to analysis of instability between two Newtonian fluids or a combination of Newtonian and non-Newtonian. Main objective of the current study is to analyze the instability in miscible flows of Newtonian and non-Newtonian fluids in porous media with anisotropic properties. In this investigation, an exploration on miscible Newtonian and non-Newtonian fluids in a two-dimensional Hele-Shaw cell is carried out. In order to analyze the effects of non-Newtonian fluid, a Carreau-Yasuda model that shows a suitable agreement with the polymeric fluids is used. In this model two dimensionless parameters of D_e and n are used to simulate the rheological behavior of non-Newtonian fluids. Three different flows are investigated here; Newtonian displaced by Newtonian fluids, Newtonian fluids displaced by no-Newtonian fluids and non-Newtonian fluids displaced by Newtonian fluids. These displacements in porous media with an-isotropic permeability, and dispersion tensor are studied to investigate the creation and propagation of the fingers. Also linear stability analyze and non-linear simulation are investigated in this study. In linear stability analyze with the quasi-steady state approximation and shooting method, growth rate in different wave numbers with presented parameters is investigated. In non-linear simulation section with accurate spectral method and taking advantages of Hartley transform, some mechanisms of propagations of fingers are analyzed in different time sequence Furthermore, an investigation on transversely average concentration, mixing length and sweep efficiency field is carried out.