SMS Backup and Restore

Any app that fits this description in the market?

Try to use http://smslemon.com. This is app lets you manage your smses by the website. All your smses are copied from phone to the service (smslemon). You can delete your smses from the phone and store on the website only. If you want to restore smses on phone back you can do it also.

You can try VeryAndroid SMS Backup. It is a smart andriod sms backup software that can copy and backup sms on Android phone to computer, restore sms backup file to any android phone, and also send messages or chat with a friend on computer directly, exactly the same as that you do on your android phone.

Good luck.

SMS2Text lets you back them up to a text file, but Im not sure about restoring. Since Im rooted Titanium Backup always backs them up and restores them as long as I have made an app data backup.

Try SMS Backup And Restore
Google Play link: https://play.google.com/store/apps/details?id=com.mdroidapps.smsbackuprestore

Features:

Backup and restore your SMS messages
Schedule automatic backups or manually back up your SMS messages
Export to sdcard your SMS messages in HTML format or export all backups to a file which you can import later
Search the SMS messages through your backups
View backup contents
Export to HTML selected conversations or restore them
Delete all SMS Messages on the Phone
Delete existing backups

SMS Backuperstore is only for Android and This software can take backup only for SMS.