One of the oldest 3D chess games was invented by the Russian mathematician Ervand Georges Kogbetlianz in 1918, and we can consider that most modern invented 3D chess games are simply variants of this game. In summary, E. G. Kogbetlianz's chess game is as follows:
The 3D chessboard is a board of six levels (I to VI) of 8x8 square chessboards, one over the other along the z-axis direction. Two players oppose each other, and each has 9 types of pieces: Archbishop (Ar), Bishop (B), Favorite, Fool (f), King (K), Knight (Kn), Queen (Q), Pawn (P) and Rook (R). There are 40 pawns, and 24 four pieces involved on each side. The pieces move and capture along the board of the fixed chessboards according to specific rules. E. G. Kogbetlianz's 3D chess game presents many inconveniences:
In this article, I have invented a three dimensional chess game which is simple to play and has the flexibility to be converted from one coordinate system to another one and vice-versa, and there is no need to use fixed level chessboards or to add chessboards along the z-direction when the pieces move higher along the z-axis direction.
My 3D chess game is as follows: There is only one 8x8 squares chessboard with the usual initial arrangement of pieces which will represent the 2D xy-plane. However, each one of its 64 squares comprises N nested (or stacked) cubes (representing levels along the z-axis) one sliding inside the other from top to bottom such that they can be elevated in high or lowered down to a specific level along the z-axis direction (see figure 1). In order to elevate or lower down the nested cubes in the 3D space, all we need is to consider magnetized surfaces of the 64 squares of the chessboard, and magnetized bottom surfaces of the pieces. In this way a pull up of a piece standing on the surface of the square of the chessboard can magnetically elevate the nested cubes to a desired level along the z-axis. On the other hand, a piece's gentle pressure on the surface of the upper level of the surface the top cube of the nested cubes will lower them down to a desired level. This will make the levels variables and not fixed. For this reason, I named my 3D chess game an 8x8xN 3D chess game, because the levels will vary from 0 to a desired level number N, (where N is an integer) along the z-axis direction. By convention the 8x8 squares chessboard or the 2D xy plane will have the 0th level. In this way one can view this 3D chess game as an 8x8x0, 8x8x1, 8x8x2, etc...
For example, if by agreement both players decide to fix the levels N to a maximum of N=6 then the game can be seen as a variable chess game (8x8x0,8x8x1, 8x8x2, 8x8x3, 8x8x4, 8x8x5, 8x8x6), and during the game the piece moves and captures will vary from the level 0 to the 6th level in the three dimensional space. Remark: The more the levels, the more the levels are higher and the more the game becomes complex and impractical. When N tends to infinity, the levels go to infinity (this will belong to the science fiction or mathematic domain.). The pieces will move from the zero to the infinite level and vice-versa. In this ways we are playing an 8x8xX 3D chess game, where the sign X is for infinity. The smaller N is (N between 0 and 5) the more the game is simple and practically playable. Using this approach, there is no need to consider fixed level chessboards or to add more fixed chessboards (as did E. G. Kogbetlianz and his followers).
In my 8x8xN 3D chess game, two players oppose each other. One with white pieces and the other with black pieces. Each one has 2 Rooks (R), 2 Knights (N), 2 Bishops (B), 1 Queen, 1 King and 8 pawns as for the usual 8x8 chess. In order to record played 3D chess games, we will use the following algebraic 3D chess notation:
jAi
where A=a,b,c,d,e,f,g,h (x-axis) , i=1,2,3,4,5,6,7,8 (y-axis) and j will vary from 0 to N (z-axis), where N is the level number, N is a positive integer. The Pawn (P) moves and captures like the usual Pawn on the level 0, (the 2D xy plane chessboard). However, in the 3D space, the Pawn can either move one level up down, left or right, or forwards, but not backwards. Example: lets say that white pawns is on the surface of the cube level 2 of the square e4, and there are black pawns resp on 1d4, 1f4, and on 1e5. Then 2e4 x 1d4, or 2e4 x1f4, or 2e4 x 1e5. The Pawns prise en passant is valid on the 2D xy plane chessboard or on the level 0, but in the 3D space it is as follows: let's say that white pawn is on 2e4 and black pawn is on 1f4. Then if on the next move black pawn moves to 3f4, then the white pawn can capture it en passant and moves to e4.
At the start of the game the pawn has the option of moving two squares
forwards on the level 0, or up in the 3D space. 
The Knight (N) moves and captures like the usual Knight on the level 0 (that is on the 8x8 square chessboard). However in the 3D space he can either move and capture 2 levels up or down and one left, right, forwards or backwards and vice-versa.
Example: N2e4 -1f4 or N2e4- 1d4 or N2e4 - 1e3 (see figure 2).
The Bishop moves and captures diagonally on the level 0. However in the 3D space, he can move up or down one or several levels along the z-axis, and he captures pieces on his file if it is free. Example: Bg1-5g1 or B5g1 x 3g6 means that white Bishop one the square g1 moves up 5 levels above that square, then he captures a pawn on the 3rd level above the square g6, because the pawn was on the same file g.
The Rook (R) moves and captures like the usual Rook on the 8x8 squares chessboard. However in the 3D space, he can move and capture up down left, right, front or backwards provided that the opponent piece is on its file, and the file is free. Example: Let's say that white Rook is on 4e4, that is, he is on the 4th level above the square e4, and black piece is on 2h4 the 2nd level above the square h4. Then on the next move R4e4 x R2h4 because e4 and h4 are on the same row of the 8x8 chessboard. So, the Rook went down two levels from 4e4 to 2h4 and captured black Rook, provided that the row is free and no other piece makes obstacle.
The Queen moves and captures either like the Rook or the Bishop on the 3D space and on the 8x8 square chessboard.
The King moves and capture like the Queen one square at a time on the level 0, and one level up, down, left, right, backwards or front in the 3D space. The castlings are like those of the usual chess on the 8x8 square chess. But in the 3D space, there is no need for castlings.
One interesting point of this new chess game is that each pieces position in the 3D space constitutes a surface in the 3D space. So a whole played 3D chess game with k moves, k greater than 1 will constitute a set of k surfaces. Moreover, in this game one can easily switch from 2D to 3D chess game and vice versa. This game has also the flexibility to be converted from one 3D coordinates system to another 3D coordinate system (ex: rectangular, cylindrical, spherical, etc...). Finally this game fits the mathematical system coordinates approach and there is no need to consider fixed levels chessboards as did E. G. Kogbetlianz and his followers.