At sufficiently high Reynolds number, all surfaces are rough, and roughness affects most flows in engineering and the natural sciences. Examples range from atmospheric boundary layers over buildings and canopies, to engineering surfaces with erosion, deposits, etc. To study the roughness effects, we take a high-resolution approach to capture the flow around individual roughness elements using direct and large-eddy simulations (DNS and LES); the goal is to elucidate phenomena which have been difficult to access using physical experiments, and to help develop engineering correlations and models. First, most experiments and turbulence models are based on a standardized type of roughness, sand-grain roughness, which can be described using a single length scale. The relationship between the geometry of an arbitrary surface and the canonical one must be known, to predict critical flow parameters such as the drag, using either experimental correlations or turbulence models. Using numerical experiments, we relate this length-scale to the roughness geometry, and propose a guideline for its prediction in the industrial setting. Next, to explain the dependence of drag on the topographical details, we examine the role of the wake of the roughness elements in the drag generation of a rough surface. The wake field is found to promote vertical momentum transfer and near-wall instability; it might provide a link between geometry details and the engineering modeling of roughness effects. Lastly, we focus on a more realistic flow scenario -- the one with freestream accelerations -- and study the combined effects of roughness and acceleration, a phenomenon widely present in engineering flows over airfoils or complex landscapes. It is first shown, by comparing equilibrium accelerating flows obtained in the present study with the non-equilibrium flows in the literature, that the roughness and acceleration effects are interdependent and depend on the flow equilibrity. Then, using DNS data of a spatially developing flat-plate boundary layer, it is found that the effect coupling develops as the roughness affects the turbulence time scale and thus the flow susceptibility of the acceleration stabilization, while acceleration changes the wake velocity and ultimately the roughness destabilization level.