There have been myriad third party solutions for doing backup of SMS on windows phone but there have not been one from Microsoft. The Redmond giant has released a free application that allows you to not only backup SMS and MMS but also lets you backup contacts on windows phone by pressing just few buttons.
All of your contacts are backed up in a separate backup+restore folder of your SD card or internal storage (in case of non-micro SD card devices). Attachments of MMS messages are also backed up.
The contacts are stored in commonly used VCF format while the SMS messages are backed up using XML format. Messages of only the first SIM are backed up. This should not be an issue to those who have single SIM devices though.
Steps to backup SMS and Contacts on windows phone
1. Install the app from here.
2. You will find it in the Settings menu. Tap on it.
3. Click on the backup button as shown above. In the next screen, select what all you need to backup.
4. Check all checkboxes if you want to backup all contacts, sms and mms messages.
5. You can backup and restore in same way.