Being able to save your messages and keep them stored on your computer is important for me as well. This would allow you to keep a log of all the messages that were sent and received. With regard to saving your messages and downloading them to your computer there are different options for you to achieve this. You can use iTunes as a way to back up your information on your computer. Here is a link which explains the steps How to back up your iPhone, iPad, and iPod touch - Apple Support . Also, you can download an application which allows you to export your messages to your computer. This is a third party application that we do not own so if you were to use this application we are not liable for any issues or problems that may occur. Here is a link to review Export Messages - Save Print Backup Recover Text SMS iMessages on the App Store. Please let us know if this was helpful for you.