Everything started approximately 15 hours ago, prior to Emerald Nightmare's raid release. Blizzard servers were attacked by DDoS (Distributed Denial of Service) attacks which rendered their services unusable. These attacks made World of Warcraft unplayable, since they caused immense in-game lags and disconnects. During said time, players weren't able to reconnect.
The First DDoS Attack
Yesterday was the release day of the initial content for Legion (September 20). World Bosses, Mythic+ dungeons and the expansion's first raid the Emerald Nightmare were opened. Servers have started lagging like crazy 15 hours ago. All the attacks continued for up to 2 hours. Blizzard reported on Twitter they're looking into the issue.
The Second DDoS Attack
Weird time for attacks but at the time of writing this article (5 am EST), the servers have again begun to be under heavy load and it seems attacks still continue strong. Symptoms are the same, in-game lag a disconnect and being unable to reconnect or being disconnected right after a successful reconnect during loading screen. It's unknown as of yet, who's responsible for the attacks. Blizzard said on Twitter they are working to resolve the issue within the shortest time possible.
If DDoS attacks will be a routine now is still unknown, we'll have more information in the next couple of days.
It's Not Only Warcraft!
This isn't just Warcraft as you might have figured out. If you look at the Battle.net desktop app, you see an attempt to connect your Battle.net ID and many players have reported similar lag issues and disconnects in both Heroes of the Storm and Hearthstone.
What Can Blizzard Do?
Blizzard alone can't do anything against the DDoS attacks. It's their service providers who get attacked. I don't get it why such a multibillion dollar company can't invest money in any sophisticated DDoS protection for their providers.