If you're someone who sits at your computer most of the day for work or frequently find yourself missing calls or incoming messages while you're at the computer, Motorola Connect might have been a feature you were happy to gain usage of. The feature is two pronged. The Motorola Connect Chrome browser extension plus Android app combination work together in tandem to provide users with a way to see not only incoming calls from you device, but also see and reply to incoming text messages as well all from your desktop so you don't miss out on anything important. Perhaps you're phone is already by you and you weren't worried about missing notifications but you rather just appreciated to ability to reply to messages from the PC because you were already using it, creating a factor of convenience.
It sounds like a rather interesting feature combination between PC and phone but the downside(for users at least)is that Motorola Connect has been down for a few days. According to reviews of the app as well as comments on the forum boards for the service, Motorola Connect seems to have gone into a dormant unresponsive status around Friday or so, and as of right now it's still not working with no explanation as to why. Droid Life has reportedly reached out to Motorola to see what the issue could be with Motorola Connect yet has not received a response from the team.
Users have apparently tried everything including your standard and uninstall and re-install methods and nothing has worked. While this is certainly a downside for those who have used the service(Motorola Connect is limited to device support on the Moto X, Droid Ultra, Droid Maxx, and Droid Mini), the good news is that in the meantime there are similar alternatives to Motorola Connect. One that I am personally using and am rather fond of at the moment is Pushbullet, which like the name suggests pushes notifications from your selected device to your computer. It'll require the Chrome browser extension as well as the app install, and you'll see any incoming notifications from your device and not just calls and text messages. Responding to messages with Pushbullet is a little different, as it doesn't allow you to respond as a true SMS to a contact number, but if you add a friend or family members email and name as a Pushbullet contact, you can push messages to their device. It seems they would have to have the Pushbullet app installed on their phone to receive messages and reply back from their device.