See here.
http://www.chess.com/forum/view/community/chrome-browser-adobe-flash-player-support-ending
Google already has it's own embedded PPAPI flash plugin named ''pepperflash''. It is shipped with the browser and updated automatically.
Google and Mozilla are both in the process of phasing out the NPAPI architecture for plugins considered outdated. Google has gone a step further and JAVA, Silverlight, Adobe Flash NPAPI no longer work with the browser at all. Firefox is not quite ready.
Since Mozilla didn't partner with Adobe for implementing pepperflash (they choosed to work on their own open source standard instead) they have kept the compatibility.
Hi, I am gonna post this here since it's the more recent thread.
Update: As of September 1, NPAPI is officially no longer supported in Chrome, https://support.google.com/chrome/answer/6213033?hl=en.
I haven't checked much on the site but appears that the Play vs Computer mode, which uses java, no longer works in Chrome. Others like the daily puzzle and Tactics Trainer still work for me though.
Google say that the NPAPI plugins are problematic (Flash certainly crashes more often than it ought!) and also have security issues and, therefore, they'll shortly not be allowing their use on Google Chrome. Chess.com uses Flash which means, as it stands, that I will have to find another engine to play my games. Since Chrome is, in my opinion, by far the most reliable engine, I don'y want to do this! I assume Chess.com is aware of this. What, if anything, is going to be done?