In a medium heavy skillet, toast walnuts over medium heat for 5 to 7 minutes or until you can smell the aroma of the walnuts, shaking the skillet once or twice during toasting. (Be attentive, as nuts burn quickly.) Remove skillet from heat; let walnuts cool for a few minutes.
In a food processor or blender, combine walnuts, Parmesan and garlic. Cover and process or blend with several on/off turns until walnuts are coarsely chopped. Add arugula. Cover and process or blend with several on/off turns until leaves are coarsely chopped.
With the machine running, gradually add olive oil and process or blend to the consistency of soft butter. (If you prefer a smooth, thin pesto, add more oil. If you want a thicker, chunkier pesto, add a little less oil.) Season to taste with salt.
Refrigerate up to 2 days or freeze up to 3 months. (Divide mixture into 1/3-cup portions before freezing, so you only have to thaw as much as you need.)