Copyright (c) 2000 Dan Troyka

This game is played with the following setup:

HOME ZONE - The set of cells where the stones of each player start.
STONES - Each stone may:
Move - Stones may slide orthogonally any number of empty cells.
Push - A stone may push a stone (of either color and not necessarily adjacent) until a wall is reached. If the stones being pushed are initially spaced out, they will be compressed together when pushed.
Swap - A stone can also swap with an adjacent enemy piece.
KO rule - A stone not push or swap an enemy stone that moved on the previous turn or is in the player's own home zone (not the stone's home zone). Such a piece is considered "fixed." 
There is no capturing.
GOAL -  Wins the player who moves all of his stones into the opponent's home zone.
A players wins also by trapping an opponent stone in its home zone (this occurs when a player places two stones in the opponent's home zone on both sides of an opponent stone in a corner. The trapped stone is completely immobilized).

From the author: Pompeii is a game of pushing, swapping, and positional attainment played on a variety of board designs that emphasize corners and "choke points." [...] The game presents many tactical choices. You can: double up your pieces so that a single move pushes multiple friendly pieces; occupy corners and choke points that impede your opponent's progress; move a piece quickly into your opponent's home zone, where the fixity of the piece creates an obstacle; or quickly move all of your pieces out of your home zone, leaving occupation of your opponent's home zone for the endgame. The appropriate tactics may vary according to the board design.

An pushing example

White just played c4-c5. Black can push c3 to cells c1 or c2 (moving itself to c2 or c3). Then White can move to d5 pushing the black stone at d5 to e5, or even make a double push, moving to e5, pushing the two black stones to f5 and g5.

Winning with a swapping trap.

Black to move. He can swap a3 with a2, and so trap the a1 stone, winning the game.

There is a ZRF to play Pompeii with Zillions. Other setups are possible, Zillions come with some variants. Here are some: