A multi-cylinder internal combustion engine is provided with a system for the variable actuation of the intake valves. At least one part of the engine cylinders is deactivated, cutting off fuel supply to said cylinders, under operating conditions that do not require the maximum power of the engine and in which one wants to reduce fuel consumption. The intake valves of the deactivated cylinders are kept at least partly open during at least one part of the discharge stages in the deactivated cylinders, hence, in the deactivated cylinders, part of the burnt gases generated during the operation prior to the deactivation flows into the respective intake conduits during the discharge stage of each cylinder. The intake valves are closed after the discharge stage. The intake valves of the deactivated cylinders are further kept closed during the compression and expansion stages in each deactivated cylinder.