Apolsoft Android SMS Transfer for Mac is an application that lets you back up all your SMSs from your Android device to your Mac. There is both a portable and an installable version or the app, but both of these versions require you to install a separate application on your Android phone. The downloadable ZIP files come with the APK file that you need to install on your phone. In my case, my phone was unable to install it because it didn't allow applications from outside the Play Store, but the Android app can be downloaded from the Store as well, so that was what I did. From the Android application, you can either back up your messages or restore them. When you back up your messages, you will create a .DB file that you need to open in the Mac app if you want to see your messages.
Android SMS Transfer for Mac will display all your received and sent messages, your drafts, messages in your outbox and those that failed to dispatch. It will also display the numbers of the contacts that you have texted. There is a nice edit mode that lets you open and modify any message that you have sent or received.
In conclusion, this application works well, but it is a bit complicated to install, especially for someone who doesn't know much about Android phones. Backing up SMS messages with an iPhone and similar tools by other developers is a much easier task, and the iPhone is a notoriously closed device.