Abstarct: Fluid flow in a curved duct is one of the main and most important flows in fluid mechanics, as it has wide applications in industry and medicine. This topic has been widely studied by experimental, numerical, and analytical approaches. It should be indicated that most of research so far is limited to Newtonian fluids, with few considering non-Newtonian fluids. In this research, the viscoelastic flow and heat transfer in stationary and rotating curved ducts with square and rectangular cross section has been investigated. The main purpose of this research is investigation of the effect of viscoelastic properties on fluid flow. Therefore, Criminale–Eriksen–Filbey (CEF) model has been used as the constitutive equation. This equation is able to model the non-linear viscometric functions especially the first and second normal stress differences. Here, the flow and heat transfer has been studied numerically and analytically. The analytical relation of force equilibrium in the core region of fluid flow in curved duct is presented using the order of Magnitude technique which is useful for understanding the mechanism of centrifugal forces resulted from curvature and rotation, Coriolis force and normal stress differences on the generating of secondary flows. Also, the perturbation method is used to investigate the reverse effect of the relaxation and retardation times of viscoelastic fluid on the flow rate in curved ducts. This method has been applied for curved circular pipe due to the some major mathematical problems for other shapes of cross section. According to these analytical results, the flow resistance ratio of viscoelastic fluids with large relaxation time is more than Newtonian fluids while the effect of retardation time of viscoelastic fluid on this flow is reverse and by increasing this time constant, the drag of curved duct is decreased. Furthermore, numerical resultants of fluid flow in curved duct with rectangular cross section implicate these phenomena. In this research, finite difference method has been used to discrete governing equations on the staggered mesh and allocating method of flow and heat transfer parameters on this mesh is according to the Marker and Cell method. Artificial compressibility method has been used to estimate the pressure in time steps of numerical simulation. Here, some numerical techniques have been applied to stabilize the numerical solution for large value of elastic properties. The validity of numerical results is evaluated and independency of these results from mesh is studied. Also, effect of parameters such as Reynolds number, Dean Number, Roseby number, curvature ratio, Elastic Number, Weissenberg Number, aspect ratio and effect of viscosity, first and second normal stress differences on the flow field and heat transfer in creeping and inertial flow (in stable and unstable situations) has been studied numerically. It has been shown that unlike Newtonian creeping flow in curved duct, viscoelastic creeping flow in curved duct can be unstable. One of the main results of this research is study of normal stress differences and its effect on flow field separately. According to this study, by increasing the first normal stress difference, secondary flow intensity will be raised. This issue can be created instability and increasing the second normal stress difference has inverse effect and can stabilize the flow.