[whpalmer4]

All 3 versions of the app support repeating projects, just as they do repeating actions. Set your project to repeat, put a start date on the project, and you're off to the races. The start date keeps the next copy of the project from starting immediately after you finish the last one, assuming you don't want that. A project to mow the lawn on Saturday morning every week might look something like this:

Mow the lawn (Start: May 4, 2013 10:00 AM, Repeat: Every week, Sequential)
Pick up sticks and other debris
Cut grass
Trim around trees
Wipe mower blades clean

If you've got the Mac app, there's an option in the Inspector to set the selected project(s) to automatically complete themselves when all of the actions are completed. The iOS apps heed that setting, but unfortunately don't offer the user interface to inspect or set it. With the settings mentioned above, when you complete the project, a new copy of the project will be created to start 1 week after the previous one did, and you won't see those actions for the upcoming project until the start date arrives, if you are showing Next Actions or Available.

To clear up a small misconception, completed tasks are NOT erased. Instead, they simply aren't shown unless you ask to see them. The eyeball icon allows you to see Next Actions, Available, Remaining, or All, and only the final option includes completed tasks and projects. On the Mac, there's a similar setting in the View Bar.

You may notice that there are multiple options for repeating. Repeat every gets you a repeat that is fixed to the calendar, regardless of when you complete the project. Paying the rent or mortgage would be a good example of this — the landlord or bank will want the next payment on time, regardless of when you made the last one. There are also options for Start again and Due again, which base the new dates on when you complete the previous copy of the project. Getting a haircut might be a good example here — if you skip getting one for a few weeks, there's no need to "catch up" and have two in quick succession.