Mozilla makes the move to a new computer easy through its Sync feature, which automatically syncs your data, including passwords and history, to an online account. Furthermore, syncing occurs between any computer that logs into Sync with the same credentials. This not only transfers your data to a new computer but also enables you to work within Firefox on any device and have your changes globally reflected.
Click the Firefox menu, indicated by three horizontal lines on the toolbar, select "Sign in to Sync" and then choose "Get Started." Alternatively, enter "about:accounts" without quotes in the address bar and click "Get Started."
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Enter a valid email address, choose a password and select the year of your birth. Check the "Choose What to Sync" box for the option to selectively sync data; otherwise, Firefox syncs everything, including tabs, bookmarks, passwords, history, desktop add-ons and desktop preferences. Click "Next" to continue.
Check your email and then click the link in the verification email you received from Firefox Accounts. Doing so verifies your account and opens the Sync Selection dialog, assuming you previously selected the "Choose What to Sync" option. Check the "Passwords" and "History" options -- and any other data you want synced -- and then click "Start." If you previously decided not to selectively sync, syncing begins automatically after verification. You can tell when syncing occurs by looking at the animated icon next to your account email address on the Firefox drop-down menu. When the animation stops, syncing is complete.
Click the Firefox menu on the new computer, select "Sign in to Sync," click "Get Started" and then choose "Already Have an Account? Sign in." Enter your email address and password to copy the synced data to the new computer.