Microsoft is rolling out a big new feature for Cortana today, allowing users to link their Google accounts. At the moment, the feature only seems to be live on Windows 10, although you should see it on iOS and Android soon enough.
You can link your account by opening Cortana, going to the Notebook tab, and choosing to add a service under Connected Services. It will be displayed as Gmail, although you can use it to manage other Google services, like Calendar. Unfortunately, you can only add one Google account at this time, so if you're like me, you'll have to choose between your work and personal accounts.
Some commands seem to work, while others don't. You can ask Cortana what events are next on your Google Calendar, but the commands don't seem to be available yet for Gmail, as first reported by Thurrott. Obviously, more functionality should be coming soon.
In fact, Microsoft hasn't even officially announced this yet, so we should find out more soon. Naturally, we've reached out to Microsoft to see if it has more to share.