Your Windows Phone backs up your text messages, but WP Message Backup is the first app to give you detailed access to those messages. In this beta version of the app, you can browse and export all of the messages you’ve sent or received. It supports normal SMS text messages, picture messages, group messages, video message, and pretty much anything else. You can export messages and conversations as plain text, CSV, or XML.

WP Message Backup does not allow you to send text messages.
A Microsoft Account with an associated Windows Phone is required, and that phone must be set to backup text messages (to configure this, look under Backup in Settings).

Note: WP Message Backup is not created by Microsoft or associated with Microsoft in any way.