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Cell migration is an ubiquitous process in life that is mainly triggered by the dynamics of the actin cytoskeleton and therefore is driven by both mechanical properties and biochemical processes. It is a multistep process essential for mammalian organisms and is closely linked to development, cancer invasion and metastasis formation, wound healing, immune response, tissue differentiation and regeneration, and inflammation. Experimental, theoretical and computational studies have been key to elucidate the mechanisms underlying cell migration. On one hand, rapid advances in experimental techniques allow for detailed experimental measurements of cell migration pathways, while, on the other, computational approaches allow for the modelling, analysis and understanding of such observations. Here, we present a computational framework coupling mechanical properties with biochemical processes to model two–dimensional cell migration by considering membrane and cytosolic activities that may be triggered by external cues. Our computational approach shows that the numerical implementation of the mechanobiochemical model is able to deal with fundamental characteristics such as: (i) membrane polarisation, (ii) cytosolic polarisation, and (iii) actin-dependent protrusions. This approach can be generalised to deal with single cell migration through complex non-isotropic environments, both in 2- and 3-dimensions.