In summary:
If you click on the launcher's "play" button, which goes gray for a few seconds and then gets re-enabled again, and the game itself is not starting (the "play" button re-enables itself when the launcher detects that the game proper has closed), try clicking on "
manage" and "
validate files". This often fixes the problem.
Note: When you do this, it may sometimes look like it's stuck at 0% (or sometimes at 100%), but this is usually normal, and just a design flaw in the launcher code. Just wait until it's finished. It may take some minutes, but usually will get there in the end, even if it looks like it's doing nothing. (The same is true when you click on the "update" button when it has replaced the "play" button.) Shouldn't usually take more than about 5 minutes, though. Maybe 10, depending on how many files it has downloaded.
If, however, you are getting the error "
incorrect manifest checksum" when you do that, then try deleting the temporary downloaded files in the Windows internet cache, and then try the "validate files" button in the launcher again.
To do that, search for "internet options" in the search bar of the Windows taskbar, and in the dialog that opens up select "delete..." and then "delete":
Note: Some people are reporting the game still not launching even after meticulously having done all that. Unfortunately I don't have any further knowledge about this. (For me these have always helped in the past, when this has happened.)