Computational screening methods have been accelerating discovery of new materials and deployment of technologies based on them in many areas from batteries and alloys to photovoltaics and separation processes. In this review, we focus on post-combustion carbon capture using adsorption in porous materials. Prompted by unprecedented developments in material science, researchers in material engineering, molecular simulations, and process modelling have been interested in finding the best materials for carbon capture using energy efficient pressure-swing adsorption processes. Recent efforts have been directed towards development of new multiscale and performance-based screening workflows where we are able to go from the atomistic structure of an adsorbent to its equilibrium and transport properties for gas adsorption, and eventually to its separation performance in the actual process. The objective of this article is to review the current status of these emerging approaches, explain their significance for materials screening, while at the same time highlighting the existing pitfalls and challenges that limit their application in practice and industry. It is also the intention of this review to encourage cross-disciplinary collaborations for the development of more advanced screening methodologies. For this specific reason, we undertake an additional task of compiling and introducing all the elements that are needed for the development and operation of the performance-based screening workflows, including information about available materials databases, state-of-the-art molecular simulation and process modelling tools and methods, and the full list of data and parameters required for each stage.