There are many tournament systems that can be used in place of McMahon. Some of them are listed here, in approximate order of usefulness. You may use any that you like; but if you have no strong preferences, we recommend that you just use McMahon. If you use a system other than McMahon, then this must be published in the literature.
Any of these systems can be played with or without handicaps. If handicaps are used, then the method of determining the handicap should be specified. For example "MMS-2" is often used in Europe and "Grade difference -1" is sometimes used in Swiss tournaments here.
Appendix D  has two useful tables specifying handicaps and komi for running small-board handicap tournaments.
This is best for 4, 6 or 8 players with number of rounds 1 less than the player count. Players are seated opposite each other at a long table for round 1. One player is the pivot and remains seated whilst the others circulate clockwise at each round.
Colours can be assigned so that players have optimal alternation. Two of the players get perfect alternation and the remainder have just one instance where two games in a row are the same colour.
Appendix D  has details on optimal colour assignment as well as a useful result sheet and table diagram for running round-robin tournaments.
All players start equal, and in each round players with the same number of wins play each other.
This is the ideal system for an even game tournament in which there are too many players for an all-play-all. Details of organisation are exactly as for the McMahon system described in sections 4 and 5. (The McMahon system can be thought of as a generalised Swiss system, or the Swiss thought of as a McMahon with everyone starting above the bar.)
Ties at the end of the tournament can be resolved either by Sum of Opponents' Scores (SOS) or by Cumulative Sum of Wins (CUSP). Neither of these methods is completely satisfactory, and playoff games should be used for important places if time permits.
The idea here is that nobody is eliminated; after each round players with exactly the same sequence of results are matched together.
This system ensures that everybody gets plenty of games against roughly equal opposition, and can be used to arrange all the players in order, though ordering is pretty arbitrary, especially around the middle of the list.
The usual ordering system is to give the losing finalist 2nd place, the losing semi-finalists 3rd and 4th, the losing quarter-finalists 5th to 8th etc., but this method puts a high premium on winning early – in a 32 player tournament the player placed 8th has won 2 out of 5 games, while those placed 9th and 17th have 4 out of 5.
All players play Swiss except the top 8 or 16 who play a knockout to determine the winner; the losers return to the Swiss section. This is best if the non-knockout part of the event uses handicaps.
This is one of the easiest types of tournament to organise, but you do need 2 N players for a tournament with N rounds. The advantages are that it produces a unique winner in the smallest possible number of games, and that games in each round can be started as soon as the players have finished their previous game. There are various disadvantages:
For a knockout tournament, the seeding system is particularly important. It is probably best to seed them so that if they win all their early games, the eight strongest players can all reach the quarter-finals, the four strongest the semifinals, and the two strongest the finals.
Appendix D  has a sheets for running 8- and 16-player knock-out tournaments.
You need an even number of teams each with the same number of players. Teams meet only once.
If there are only two teams, say A and B, the pairing can become messy if you are not systematic. The following method was devised by Mr S Niwa of the Nippon Club.
For round 1, seat players on a long table with all team members of the same team on the same side, and playing the same colour. From round 2 on team A remains fixed. The players of team B shufffle down 1 place, and the player who drops off the far end of the table cycles round to board 1. The teams as a whole swap colours each round.
For more than two teams you should play all teams against each other. Since teams meet only once, players can be paired as you like at each round.