-- Optional: additional extra virgin olive oil (for storage)
Place the basil leaves and garlic in a food processor or blender, and mix well. Add the walnuts, and continue to blend until the nuts are finely ground.
Keep the machine running as you drizzle in the olive oil. When you have a smooth paste, transfer to a bowl, and stir in the Parmesan cheese. Season to taste with salt and pepper.
If you are not going to use the pesto right away, place in a container that has a tight-fitting lid. Smooth out the top surface of the pesto, and pour on enough olive oil to cover it completely. This creates an air-proof seal that helps preserve it well. Cover tightly and store in the refrigerator.