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There are two types of storage on iPhone. One is iCloud Storage and the other is device storage (that inbuilt device storage, i.e the one came with iPhone like 16GB, 64GB or 128GB). The device storage can't be upgraded. If it is getting full, then you have to delete unwanted movies, photos, videos, musics etc from your iPhone (you can backup them to your computer first)
You can upgrade the iCloud storage if your iPhone is asking you to get iCloud storage.
Go to Settings > General and tap Storage and iCloud Usage. You will see your device storage info under Storage section and iCloud storage under iCloud section.
If you are running out of iCloud storage, then you can upgrade the storage from Settings > iCloud >Storage & Backup > Buy Storage Plan
If you are running out of Storage (i.e device storage), then you have to delete unwanted photos, videos, music etc from your iPhone. That is the only way. Open Photos and delete the photos or videos from All Photos or Camera Roll. Then open Recently Deleted album and delete from there as well.
Then open Music or Videos app and delete unwanted music, movies etc
Before deleting them, you can follow the steps below to backup them to your computer
First download and install latest iTunes on your computer from here --> http://www.apple.com/itunes/download/
To transfer photos/videos to computer :
Connect the iPhone to your computer with the USB cable. If asked to trust the computer on your iPhone, tap Trust Computer.
Open Computer or This PC (Windows explorer). You can find your iPhone in there. Right click on your iPhone's icon and select Import Photos and Videos. This will download all photos and videos to your computer's Pictures folder. So open Pictures folder and make sure they are downloaded there.
To transfer music to computer :
Open the iTunes application. Then connect your iPhone. If iTunes asked you to setup as new iPhone or Restore from Backup, choose Setup as new. Then in iTunes, click on the iPhone icon at the top left next to Music, Movies etc icon. This will bring Summary page of iPhone. Click Backup Now button and make sure "To This Computer" is selected. If asked to Transfer purchases, click Yes and this will Transfer the music to your computer.
Now you can delete them from your IPhone to save storage space.
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