Geological faults movements generating earthquakes, a vehicles' tyres rolling on the pavement, and a chalk writing on a blackboard are all different examples of frictional systems. In these systems, which are everywhere around us, two separate bodies are in contact and in relative motion, interacting with one another. Under these conditions, several phenomena arise at the interface: friction, wear, and lubrication being the main ones. Wear, in particular, is the loss of material from the surface of one (or both) the moving bodies. This phenomenon is of interest as it leads to loss of usefulness of manufactured objects and health concerns for patients with implants, to name just a few of important consequences due to material degradation. Yet, our knowledge on the topic of wear is still scattered, with many observations and models that are system dependent. How the surface morphology changes during wear processes is one of those aspects that are not well understood, and that at the same time affect significantly wear itself. Therefore, the aim of this dissertation is to investigate the role of surface roughness in wear processes upon dry sliding. The work focuses on wear of the adhesive type, as it is the most common one (together with the abrasive type). The first part of the thesis addresses the topic with numerical investigations. Two-dimensional systems are modelled with a discrete approach, where two surfaces are rubbed against one another. In this setup, a wear debris particle is eventually formed and it is constrained to roll between the sliding surfaces. It is shown that the method reproduces the evolution of the surface roughness into the self-affine morphology that is observed for different frictional surfaces. Furthermore, the interplay between surface roughness, adhesion, and wear debris particle size is investigated, and a minimum size for the debris particle is determined, based on a critical length scale recently derived for adhesive wear processes. These sets of simulations bring further observations on the wear process, like the evolution of the work due to the tangential forces and of the wear volume. The latter in particular displays an overall increase: debris particle accretion is favoured over its break down. This leads to the second part of the dissertation, where an analytical framework is presented that allows to explain the tendency of the debris particle to grow in volume, instead of depositing material onto the mating surfaces. The general approach of the work aims at uncovering underlying mechanisms of wear processes, and it is not restrained to some specific application. While on one hand this means that the work cannot cover the specificity of some frictional systems, on the other hand it leads to fundamental insights that are relevant from the nano- to the geological-scale.