We propose a computational algebra approach to some classical problems in the theory of linear partial differential equations. Our approach is based on the developments of new tools in computational algebra, as well as on the algebraization process which has occured in the theory of partial differential equations over the last 30 years. We describe several problems in well-known areas (calculation of Ext modules, free resolutions, Koszul complexes, Noetherian operators, etc.) but we also describe some new problems such as the development of a possible theory for weakly commutative systems.