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I wonder if they use a relational database like MySQL or do they opt for a NoSQL like CouchDB or MongoDB.

The Bitcoinica source code reveals that they used a MySQL database.

closed as not constructive by o0'., technology_is_overrated, Nick ODell, David Perry Apr 8 '13 at 18:00

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There are multiple Bitcoin Exchanges out there, most likely they are using different DB Systems.
To be sure you should contact the particular representatives of the Exchanges.

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