There are several different ways to create a backup of a MySQL database. The most common is what has been discussed mostly above and which is covered in the documentation mentioned. This creates a single text file that lists all the SQL commands needed to recreate the database as it was. Other methods include simply taking the actual database data files from the original server and compressing and archiving them. This is usually less than ideal because you can run into problems with differences between versions of MySQL.

Does the original database still exist somewhere, or is it only in the backup file?

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