Chess Variants Around the World

Chaturanga, Indian Chess, spread around the world and morphed into many other variants, especially in East Asia. These are in three main categories: Xiangqi type, Shogi type, and Chaturanga type, the latter including great chess. Here I'll be listing some, not all, of the historical variants.

Xiangqi 

Janggi

Xiangqi

Xiangqi is Chinese chess, played on a 9x10 board, on the intersections rather than the squares. Janggi is a modification of it played in Korea which increases the power of the pawns and removes restrictions crossing the middle of the board. This makes the game more like Shogi than Xiangqi, which is very tactical and is played more move by move than with general principles of strategy. In addition, Janggi gives more freedom to the king, making checkmate harder to achieve, but balances this out by giving the pieces more abilities within the palace.

Shogi

Shogi extends the idea of promotion (limited to pawns in Xiangqi, as well as international chess and the rest of the chaturanga-type games) to all the pieces, creating new promotion tactics and including many interesting pieces. There are a huge number of shogi variants.

1.  Games with Drops
1. Standard shogi
   This is of course the most popular form of shogi, with a huge playerbase in Japan and worldwide and a competitive professional scene. 
2. Tori Shogi (bird shogi, all the pieces are named after birds)
   This is one of my favorite variants: games are fast paced and tactical, and there is no positional play, just attack and defense. Like Xiangqi, Tori Shogi is played move by move, responding to your opponent's threats and creating your own, but it is quite different because there are no unpromoted ranging pieces besides the quails, which are rarely moved very often. 
3. Wa Shogi
   I don't know much about Wa Shogi except that it is heavily biased toward forward movement and that the pieces are all animals. 
2. Sho Shogi
   Sho shogi is in the middle of the drops games and big shogi. It is standard shogi, but without drops (it was considered a children's game until drops were added), and with the drunk elephant, a step-mover characteristic of the larger variants that promotes to a royal crown prince.
    
3. Big Shogi
1. Chu Shogi
   Considered by a lot of people to be the greatest chess-like game ever created, and I definitely share this opinion. It's well balanced (the starting position helps with this), it has powerful pieces to make up for the large board size and keep it interesting, and besides HaChu, which isn't insanely strong, there are no engines to ruin the fun (this is my opinion only, but I think engines make games too theoretical and prevent people from analysing positions without them).
   
2. Dai Shogi
   Not my favorite: It is just an extension of Chu Shogi with more stepping pieces, just making it more boring. It is however the first or third recorded form of shogi, depending on whether or not you count Heian Dai Shogi and Heian Sho Shogi.
   
3. Tenjiku Shogi
   My new favorite thing: it's wild, tactical, and fascinating even though the board is huge because the pieces are more than powerful enough to compensate. Alertness is required to play (or theoretical knowledge, and preferably both), as checkmate can be blundered as early as 3 ply in.
   
4. Dai Dai Shogi
   I'm playing a correspondence game of Dai Dai right now, and I've found that it fixes one "problem" that Maka Dai Dai Shogi has, though I'm not sure it's a problem: the fact that the opening is almost completely played with a few very powerful pieces. This game is important because it is the first game in the big shogi group of games to use the promote on first capture system. Dai Dai is a good size (17x17), and the game would be very boring if promoted pieces didn't exist early on. One other thing I noticed is that the step-movers are very well placed at the beginning of the game to make an unsupported lion-piece invasion more or less useless. The reason for this is that the ranging pieces on the back ranks would capture the lion, lion dog, or furious fiend if any igui capture was made. This situation was reached in the correspondence game.
   
5. Maka Dai Dai Shogi
   I have zero experience with this game, but something that is different than smaller variants is the king promotion by capture, to the Emperor, a piece that can move to any uncontrolled empty square. This is such a powerful promotion that one middlegame strategy is to clear a path to a weak enemy piece so that the king can capture it and promote.
   
6. Tai Shogi
7. Taikyoku Shogi
   The last two variants are just ridiculously large. Both of them together have probably only been played once or twice, but no one is sure. The only thing anyone knows is that Taikyoku Shogi was played once on a Japanese TV show.
   

Chaturanga

Besides Indian Chaturanga and Perso-Arabic Shatranj, there is also Thai Makruk and Burmese Sittuyin. Makruk and Sittuyin are very similar to each other and to Shatranj, except that Sittuyin is a placement game, with pawns split and on two ranks. I would also count Mongolian and Indian great chess variants, including Tamerlane chess, which reminds me of king of the hill sort of.