SMS Backup Export
SMS Backup Export is an android app which I released in 2014. It will let you export your phone's SMS messages from your old phone and import it to your new android phone. The exported file can also be used as a backup file from your messages on same phone.
The other feature of this app is exporting the messages as a single HTML file. The exported HTML file will have built-in features like searching for messages from a particular contact, or with a particular text in them.
This app also supports unicode characters which makes it able to deal safely with non-english characters.