This work explores mechanisms for pattern formation through coupled bulk-surface partial differential equations of reaction-diffusion type. Reaction-diffusion systems posed both in the bulk and on the surface on stationary volumes are coupled through linear Robin-type boundary conditions. The presented work in this paper studies the case of non-linear reactions in the bulk and surface respectively. For the investigated system is non-dimensionalised and rigorous linear stability analysis is carried out to determine the necessary and sufficient conditions for pattern formation. Appropriate parameter spaces are generated from which model parameters are selected. To exhibit pattern formation, a coupled bulk-surface nite element method is developed and implemented. The numerical algorithm is implemented using an open source software package known as a deal.II and show computational results on spherical and cuboid domains. Also, theoretical predictions of the linear stability analysis are verified and supported by numerical simulations. The results show that non-linear reactions in the bulk and surface generate patterns everywhere.