The fruit of a deciduous tree (also called noce), consisting of an external hull (mallo) surrounding a hard shell (guscio), which contains the edible seed (gheriglio).
My aunt Lucia used chopped walnuts (noci) and egg whites (chiare d'uovo) to make croccanti for Christmas, then delivered small packages of them to family members. My father remembers cracking the walnuts his sister would use. At my parents' house, a tray of walnuts and hazelnuts (nocciole) was always on the kitchen counter and as a child I loved using the schiaccianoci (nutcracker) to extract the gherigli from their shell.
A bottle of home-made nocino could always be found in the glass cabinet in my aunt's kitchen. Nocino is a dark brown liqueur, whose recipe includes whole walnuts still with their green hull, cinnamon, cloves and lemon peel infused in alcohol. My aunt prescribed a small quantity of it to aid digestion and to cure a number of ailments.
The cutest Italian expression involving walnuts is un guscio di noce (a walnut shell), used to describe a tiny boat.
We also use noce in the name of fruits that are shaped like a whole walnuts, e.g., noce moscata (nutmeg) and noce di cocco (coconut). When a recipe calls for a small piece of butter it may say una noce di burro.