Interest in microfluidics has increased dramatically in recent years, with applications spanning a wide range of fields. However, despite several advances, design of microfluidic devices still relies largely on trial-and-error. This thesis aims to go beyond this approach in favour of a rational design of microfluidic devices based on theoretical and numerical design rules and algorithms. More specifically, this research focuses on understanding and controlling fluid dynamics in applications involving complex non-Newtonian fluids in shear and extensional flows. Biomimetic principles and shape optimisation methods are employed to propose new designs for single-phase fluid flow. Furthermore, the single-phase numerical solver is extended to cope with two-phase systems, thus paving the way for new applications of these techniques. Focusing on shear-flows, a biomimetic principle appropriate for fully developed flows has been extended here to be applicable for non-Newtonian fluids, described by the power-law constitutive relationship. The derivation of the principle leads to a biomimetic rule that provides the appropriate dimensions for designing customised microfluidic bifurcating networks, able to generate specific wall shear-stress gradients along consecutive generations. A range of power-law fluids is examined numerically demonstrating great agreement with theoretical predictions. In terms of extensional flow, a range of shapes are proposed for designing microfluidic channels for studies related to the response of complex fluid systems under homogeneous strain-rate. Optimisation techniques are employed for finding the appropriate shapes to generate homogeneous extensional flows along the flow centre line of single stream (contraction-expansion channels) and the multi-stream designs (T-channels and flow focusing devices). The optimised geometries proposed exhibit enhanced performance compared to well defined geometrical shapes. The in-house single phase solver used in all numerical studies is upgraded here in order to solve numerically 3D-problems related to two-phase systems described by the Phase Field method. Here, the code is validated for 2D-problems only, using a range of test-cases demonstrating a very good quantitative agreement. Keywords: Non-Newtonian fluids, Shear-thinning and shear-thickening behaviour, Bifurcating networks, Biomimetics, Optimisation, Extensional flows, Two-phase systems.