PS3 to PS4, Xbox 360 to Xbox One
Moving your character from last-gen consoles to new-gen consoles is a bit of a pain. Here's the full process:
Export Your Save from Diablo 3
1. Launch Diablo 3: Ultimate Evil Edition.
2. Download and install patch 1.0.3 for Diablo 3.
3. Log in to your Sony Entertainment Network (SEN) or Xbox Live (XBL) account.
4. Proceed to the main menu. Navigate to "More" > "Export Save."
5. Confirm this selection and proceed through any prompts that appear.
Import Your Save to Ultimate Evil Edition:
1. Log in to the same SEN or XBLaccount from which you exported your save.
2. Launch Diablo 3: Ultimate Evil Edition.
Once you've done that, the game will now scan Blizzard's servers for your exported save data and load it.
Note that if you launched Diablo 3: Ultimate Evil Edition on your new platform before you exported, you have to manually import the save. It's less annoying than it sounds.