A detailed knowledge about multiphase flow systems involving non-Newtonian fluids is of extreme importance for several industrial fields. Due to its frequent occurrence, in this study particular focus is placed in a specific gas-liquid flow pattern - slug flow. A commercial CFD package with the VOF methodology was used to simulate the rise of individual Taylor bubbles through vertical columns of stagnant and flowing inelastic non-Newtonian liquids. The CFD application was first validated with in-house experimental data. The inferring of the influence of shear-thinning and shear-thickening rheology of the liquid phase in the hydrodynamics surrounding an isolated Taylor bubble was the next goal. The behaviour of the main hydrodynamic features of each flow region (around the nose, liquid film and wake) was determined, and its dependence on the liquid rheological nature was analysed. A similar procedure was then applied for systems where the liquid phase consisted on a typical shear-thickening fluid (STF) characterized by a mixed rheological behaviour. This particular aspect introduces an extra degree of complexity that is relevant when large variations of the shear rate are involved like the ones occurring in slug flow systems. (C) 2017 Elsevier B.V. All rights reserved.