For descriptions of the two main types of space chess known hitherto, those by Maack and Kobetliantz, and numerous others, I will have (for reasons of space) to refer the reader to David Pritchard`s Encyclopedia. Here I outline the general principles and follow this with a new game of my own, provisionally called Cuboid Chess.
Threedimensional 'boards' are shown by means of a set of 2dimensional boards representing successive slices through the 3dimensional region, the labels of the boards, A, B, C, ... acting as the third coordinate. The boards used hitherto have nearly always been cubes n×n×n, for some number n, or a boxshape a×b×c. Cuboid Chess however is played within a very roughly spherical volume (a 6×6×6 cube with a 4×4 area added to each face).
A 4×4×4 space chess can be played on the normal 8×8 board, the four layers A, B, C, D being the four quarters, but the chequering may be misleading.
A move in three dimensions is represented by three coordinates (r, s, t), indicating that the move is equivalent to r (wazir) steps right, s steps forward and t steps up. (Or respectively left, backward and down if the numbers are negative.) We express the 'pattern' of the move by the corresponding positive or zero values {r, s, t}. The length of an {r, s, t} move is √(r² + s² + t²).

 
D  E  

 
F  G 
Some moves of {1,2,3} X, and {0,1,2} * from the centre, Dd4, of a 7×7×7 board.
If all three numbers are different and none is zero an {r, s, t}leaper can move in 48 different directions! The simplest leaper of this type is the {1, 2, 3}leaper, or √14leaper, called a Hippogriff in Kogbetliantz`s game. If one pair of the coordinates is taken to specify a move on one of the layers of the board, then the third coordinate represents a move up or down. Thus a {1, 2, 3}leaper either moves vertically one layer and makes a {2, 3} zebra move or vertically 2 and makes a {1, 3} camel move or vertically 3 for a {1, 2} knight move, as illustrated by the marks X in the figure. In each case there are 16 choices of move (8 leaps × 2 vertical ways) whence 48. Since this piece has an even number of odd coordinates it is confined to cells of one shade in the chequering.
Fortunately when there is a zero coordinate or two coordinates are equal the number of moves to be considered reduces considerably. Any 2D move can be made in 3D space in each of the three coordinate planes passing through the cell initially occupied by the moving piece. In this way each 2D (or 1D) piece defines a corresponding Space Piece. Thus the Space Rook makes moves of type {0, 0, n} in 6 directions (updown, leftright, and tofro), where n can take any value except zero, the Space Bishop makes moves of type {0, n, n} in 12 directions and the Space Knight moves {0, 1, 2} in 24 directions (shown by the marks * in the figure).
Moves with all three coordinates nonzero are 'true' 3dimensional moves, and pieces making them are 'essentially' 3D pieces. The simplest is the Sprite, which is the {1, 1, 1}leaper, or √3leaper. Its corresponding line piece, the {n, n, n}mover, or √3rider, is called the Unicorn in Maack`s game (but 'Fool' in Kogbetliantz`s).
The Rook moves through the faces of the cubic cells, the Bishop moves through the edges of the cubes, and the Unicorn moves through the corners. Thus a Unicorn has a choice of 8 directions of movement. A Rook has access to all the cells of the board, in a series of moves, but it takes two Bishops to patrol the whole space, and four Unicorns.
The King in twodimensions can be defined either as a Wazir + Fers or as a piece that moves to every 'adjacent' cell (by which we mean any cell that has a boundary point in common with the initial cell). In three dimensions these two definitions are not equivalent. From the definition of Space Piece given above it follows that a piece that has the usual King moves in any plane is properly called a Space King. A piece that moves to every adjacent cell in space is a Wazir + Fers + Sprite which I call a Cubi(c)King. The royal pieces in the Maack and Kogbetliantz games are of this type. The CubiKing moves to all 26 outer cells of the 3×3×3 cube around it.
The choice of the CubiKing rather than the Space King as the 3D monarch is supported by the fact that one Space King could stalemate another on a cubical board (e.g. on Aa1 and Bb2) which may be considered an undesirable property for the royal piece, or at least not analogous to the 2D case.
In Cuboid Chess however I use Kings of both types.
The Space Queen, is of course Rook + Bishop (used in Kogbetliantz`s game under the name 'Favourite'). Its more powerful cousin, the CubiQueen, runs in the directions that the CubiKing walks, i.e. Rook + Bishop + Unicorn. Both types of Queen are used in my new game.
In Cuboid Space Chess there is a Royal Family of four: Space King and Queen and Cubic King and Queen. All are Royal, but the Queens are permitted to pass through check. The aim is to capture any one of the four. Thus checkmate occurs if any one of the four pieces is in check and the check cannot be parried without leaving at least one of the four in check. The Space King, being the weakest of the four is the piece most likely to be attacked, but forks and skewers can also be fatal.
This weaker type of checkmate is to counterbalance the increased mobility of all pieces in space (which, as Schmittberger has pointed out, see EVC p.310, makes normal checkmate difficult if not impossible.)
It is convenient to call the R + U a CubiRook and B + U Cu(bi)Bishop. A pawn moves like a king but forwards only (i.e. towards the opponent`s side), so we can have Space Pawns and CubiPawns, the latter having the added Sprite capture move.
Cuboid Space Chess
An impression of the Cuboidal Board in the round.

The opening position for 'Ordinary' Cuboid Chess is shown above. Two normal, but distinguishable sets are needed. The upright symbols are Space pieces and the inverted symbols Cubipieces. The Royals occupy the central cells of the 4×4 home base. The inverted Knight is the {1, 2, 3}, but other pieces can optionally be substituted.
Let us finally examine some other distinctive space piece possibilities. The 3Dmoves with coordinates non zero and less than 3 are: {1, 1, 1} {1, 1, 2}, {1, 2, 2} and {2, 2, 2}. The first and last are Unicorn moves.
The {1, 1, 2} move is of length √6, and I call the corresponding rider a Sexton, it is confined to cells of one colour like a Bishop, and can reach all the cells of that colour (PUZZLE 1: Show how the Sexton gets to a diagonally adjacent cell in fewest moves.)
The {1, 2, 2} move is of length √9 = 3. Thus the Threeleaper, becomes of special interest in Space: besides its 1D move {0, 0, 3} it has this 3D move. Unlike the Threeleaper in two dimensions, which is confined to 1 cell in every 9, in three dimensions it can get to every cell of the board! (PUZZLE 2: Show how the Threeleaper gets to the next cell in the rank in fewest moves.)
In Space we also encounter two other doublepattern fixeddistance leapers before we reach the familiar 5leaper, these are the √17leaper or Space Giraffe which moves {0, 1, 4} and {2, 2, 3} and can reach any cell, and the 3√2leaper or Space Tripper which moves {0, 3, 3} and {1,1,4}, and is confined to one colour.
In 'Advanced' Cuboid Chess the inverted R, B and N represent Threeleaper, Tripper and Giraffe, the {0, 0, 3} and {0, 3, 3} moves being blockable.
Solutions to Puzzles