This article considers a mathematical model for tumour growth based on an acid-mediated hypothesis, i.e. the assumption that tumour progression is facilitated by acidification of the region around the tumour-host interface. The resulting destruction of the normal tissue environment promotes tumour growth. We will derive and analyse a reaction–diffusion model for tumour progression that includes different aspects of pathological and healthy cell metabolism to check tumour growth. Our results can provide insights into the governing dynamics of invasive processes that may suggest possible intervention strategies, leading to the design of new and better experiments or treatments.