Why do apps I've downloaded/purchased directly on my iPhone not sync/appear in iTunes?

Many people experience difficulty getting applications they've purchased or downloaded free directly on their iPhones to appear within or sync to their applications tab in iTunes. The solution, though a bit hard to find for some, is thankfully quite simple:

1. Connect your iPhone to your computer and open iTunes.

2. Sync your iPhone (optional, but recommended).

3. Right click on your iPhone in the left hand pane of iTunes, and select "Transfer Purchases"

If your iPhone doesn't sync and/or the apps on your iPhone don't appear in iTunes' applications tab, there's one more thing you might check -- make sure you have authorized your computer to use the iTunes store. From the top menu in iTunes, click Store --> Authorize computer. Once you have authorized, repeat the steps above.