Step 1: Root your Galaxy S III by following one of the following methods based on which variant you are using: International Version GT-i9300, AT&T / T-Mobile / Sprint. Be sure to flash ClockworkMod Recovery if you haven’t already.
Step 2: Download CyanogenMod 10 for your Galaxy S III variant: International Version, AT&T, T-Mobile, Sprint [Official Forum Threads]. Place the .zip file directly under /sdcard directory on your S III for easy flashing in Step 5.
Step 2: Shut down your Galaxy S III. Press and hold Volume Up, Home and Power buttons until you boot into ClockworkMod Recovery.
Step 3: Take a NANDroid backup.
Step 4: Navigate to install zip from sdcard > choose zip from sdcard > [file you transferred to your Galaxy S III in Step 2].
WARNING! All your personal data will be deleted in this step. This is a safety requirement for whenever you flash a new ROM.
From the main menu, select wipe data / factory reset.
Step 5: From the main menu, choose reboot system now.
