An important result from Behrend-Pearce-Petkova-Zuber proved in the late 90s that solving Cardy's equation in a boundary rational conformal field theory is equivalent to finding the non-negative integer valued matrix representation (NIM-reps for short) of its Verlinde algebra, or fusion ring in mathematical terms. I will present some recent advances towards the study and classification of NIM-reps for several different families of fusion rings, and how we can use these to help us tackling further problems in category theory