By default, out of stock products remain visible but have are greyed and crossed out.
You also have the option to hide them from your web, embed and print menus. To do this, click on the "Web & Embed" button on the left hand side. Under the "Content" drop down, there will be an option to show or hide out of stock items.
Out of stock items on Displays always remain visible but are greyed and crossed out. They can be removed or replaced at any time by editing the Display.
Archiving products
If you're looking to archive products, see: How do I archive a product?