The computational modeling of fluid flow problems in engineering applications often involve material systems whose complex molecular and microscopic structure gives rise to non-trivial and, sometimes, counter intuitive rheological behavior. In this respect, and contrarily to the classical models in computational fluid dynamics, the coupled governing partial differential equations must account for complex material properties, expressed by means of constitutive equations. Additional modeling and numerical challenges are then raised, which classical computational fluid dynamics methods are unable to properly solve. Aiming to overcome these limitations, the emerging field of computational rheology focus on bridging the gap between computer simulation and fluid flow problems involving complex rheological behavior. Nevertheless, the development of mathematical models, numerical methods, and computational codes in this subject has been uneven with the ever-increasing complexity of a wide range of fluid flow problems, from both the perspective of the fundamental science and the engineering practice. Rheological constitutive equations have, generally, a partial differential nature, which results into larger non-linear systems to solve and, therefore, leads to an increased computational effort. Moreover, the computation of these constitutive equations are prone to instabilities and present poor robustness from their hyperbolic dominated nature, which deteriorate solution accuracy and convergence and, therefore, also significantly increase the demand of computational resources. Consequently, there is a growing concern that standard and outdated numerical methods, widely implemented in commercial and open-source codes used in industry, are improved or replaced by new and innovative alternatives, aiming to obtain more accurate solutions, more stable computation procedures, and significant computational gains. In that regard, the proposed research targets the investigation, development, and verification of a new class of methods for the computational rheology context. The proposed new class of methods are developed within the high-order accurate finite volume framework, where error con- vergence higher than the classical first- and second-orders are achieved under mesh refinement, therefore resulting into substantial accuracy gains. Moreover, recent and comprehensive research on this topic has demonstrated that significant gains in computational efficiency, both in terms of time and memory usage, are also achieved when higher-order accurate methods are used to obtain the same solution accuracy level given by lower-order ones. A comprehensive analysis and verification, both of the numerical develop- ments and the computational implementation, are planned to ensure that the proposed methods and tools are understood, assessed, and efficient. Also, future validation with viscoelastic fluid flow problems in the context of micro-injection moulding, with the collaboration of the industrial partners, are targeted to prove the effectiveness and usefulness of the proposed tools. Additionally, although traditional polymer processing applications, such as extrusion and injection moulding, usually require less complex con- stitutive equations, validation with problems in this context will be crucial before applying the proposed methods in more advanced and computationally demanding applications.