Backing up your text messages allows you to retrieve them in the event that the phone stops working, needs to be reformatted or is lost or stolen. You have several options for saving text messages on an Android phone, including saving to a microSD card, saving to a computer and saving to a cloud-based service like Gmail.
External Media
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If you'd like to save your text messages to external media, saving to a microSD card is an option. There are several apps in the Play Store that can be downloaded to assist with saving to an SD card, including the app SMS Backup & Restore. This app allows you to save all of your text messages in an XML file format that can be stored on your external media. You can set the app to back up once or at a set time automatically.
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Computer
You also have the option of backing up your Android text messages directly to a computer if you don't have an SD card or would rather store the files on your desktop or laptop. Wondershare MobileGo for Android Pro is a software suite that allows you to manage your smartphone from a Mac or Windows computer. You can transfer text messages to your computer by selecting the text thread in the program and clicking "Import/Export." You can also send, receive and delete text messages while your phone is connected or use SMS Backup & Restore to create an XML file and then transfer it manually to your computer.
Cloud
Using a cloud-based service to save your text messages allows you to retrieve them whenever you have an Internet connection, which can be helpful if you need to access your text messages when you travel. SMS Backup + can be downloaded from the Play Store for free. This app allows you to sync your text messages, multimedia messages and call logs directly with your Gmail account. You'll have to enable IMAP within your Gmail account and the app can only restore text messages and call logs. You'll have to log in directly to your Gmail account to access your multimedia messages.
Transfer
If you've bought a new Android smartphone you can also transfer text messages from your old device. SMS Backup & Restore, Wondershare MobileGo For Android Pro and SMS Backup+ can all be used to transfer text messages between two devices. You can also use a third-party cloud-based texting app to replace your default text messaging program, which keeps all of your conversations saved in the cloud so you can access them wherever you have an Internet connection. Your mobile carrier may also have a backup program to assist with transferring files, like the Verizon Cloud app.