Transferring contacts from an old iphone to a new one?

Hi, I backed up my iphone 4. Then I plugged in my new iphone 5 and pushed restore back up but it didn't transfer anything. It also showed the last back up as when I had first plugged in that phone. It didn't show the back up I did with my iphone 4. What have I done wrong and how can I transfer my contacts?

Added (1). When I back up my iphone 4 (on the manual back up) it shows up in my lists of back ups, but when I try to restore that back up to my iphone 5 it isn't in the list of back ups for me to choose from.

What are you backing the old phone up to? I backed my 4 to the cloud, then activated my 5, and the contacts pulled right in.

You have to transfer contacts from old phone to icloud to new phone on the computer

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