Yes, but some cost money. You can buy them on the Apple Stiore ( App Store)
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you have to have an itunes account.
32GB iPod Touch only has 28GB of free space without any other downloaded programs/apps. A single game takes up an average of 500MB, so you can download around 50-60 games, but that number will vary depending on the games you download.
Games for the iPod Nano and iPod Classic cost about $4.99 each. However, there are certain apps for the iPod Touch and iPhone that are free.
on the app store there is an area with a couple hundreds of free apps
It all depends on what size of games your playing.