Both. It's a mindsport game.
Is chess a sport or a game?
Some countries like us treat it as a sport. I like that. Although really, it is just a game. Scrabble, checkers, backgammon, game of the generals, chess and many more are just games. Chess may be the game of games but it is still a game. 
To add to that, freaking golf is not a sport!! Walking around leisurely and trying to sink a tiny ball in a hole is not a sport!! It's a hobby. Just like flying a kite!!! 
Hobbies: golf, bowling, billiards, darts, shooting, archery (unless you are shooting each other), karaoke
, etc...
According to answers.com
'Chess today is a recognized sport of the International Olympic Committee. The first official World Chess Champion, Wilhelm Steinitz, claimed his title in 1886; the current World Champion is Viswanathan Anand'
According to Wikipedia
'Chess is a recognized sport by the International Olympic Committee with FIDE being the recognized International Sports Federation for chess since June 1999. As a member of the International Olympic Committee, FIDE adheres to its rules, including controversially having doping tests. The prospects of chess becoming an Olympic sporting event at some future date remain unclear. The naming of FIDE's team championship as the "Chess Olympiad" is of historical origin and implies no connection between this event and the Olympic Games'
Does it need to be physical to be considered a sport?
Good question. I don't think the meek like us has a say on it. Media, pretty much controls it. You already know that I'm quite opinionated. In sports, I draw the line myself and not be enslaved by the ones in control. If we let all contests be called sports, spelling bee, all food-eating contests, singing contests, just to name a few will be sports. For me, sports are physical contests between two sides where one side can physically stop the other from winning. I allow exceptions, though (e g: races, track and field) Boxing is one classic example of a true sport. UFC on the other hand is too brutal for me and I just treat it as a deadly game, not a sport. It's like American Football (classic team sport) without helmet and pads. That's just my opinion and should be taken lightly. 
It depends on what definition of sport you use.
Among man definitions, one states that a sport is an activity involving physical exertion governed by a set of rules and is often taken competitively. Another definition states that it is simply an active pastime, a recreation. Chess is not a sport according to the former, but can be considered one based on the latter.
I think chess is both a sport and a game.
Mas nakakapagod mag-isip kaysa magbuhat ng mabigat hehehehehe.
What do you think of chess, a sport or a game? Why? Your opinion, please.