Cellular adhesion is a fundamental process in the development of scaffolds for tissue engineering; in the design of biosensors and in preparing antibacterial substrates. A theoretical model is presented for predicting the strength of cellular adhesion to originally inert surfaces as a function of the substrate topography, accounting for both specific (ligand-receptor) and non-specific interfacial interactions. Three regimes have been identified depending on the surface energy (gamma) of the substrate: for small gamma, any increase in roughness is detrimental to adhesion; for large gamma, an optimal roughness exists that maximizes adhesion; and for intermediate gamma, surface roughness has a minor effect on adhesion. The results presented are in qualitative agreement with several experimental observations and can capture the long-term equilibrium configuration of the system. The model proposed supports the notion for rationally designing substrates where topography and physico-chemical properties are tailored to favour cellular proliferation whilst repelling bacterial adhesion.