One of the key features of chess is the royal piece, a piece whose removal means a loss and therefore cannot be left at risk of capture. For orthodox chess this piece is the king, and chess ultimately is a game whose goal is to remove the opponent's king. In practice this finishing blow need not happen; just subjecting the king to the threat of capture with no legal reprieve is death enough.
For this essay the term royal piece will only be used regarding pieces whose capture signifies a loss in the section where it doesn't move the same way as the king. Until then king and royal piece are interchangeable.
This is checkmate, a logical step to take from how king capture is a loss. Also stemming from this rule is check, or in this case the idea that if the king is under threat of capture removing this threat becomes immediate priority(any move made not to this end is considered illegal and does not stand). Now these seem obvious, but some games do away with these formalities and make king capture the objective, disregarding these extra notions; though leaving your king vulnerable is still a dumb move there is no longer any obligation to do anything about it. This rule also affects another thing which I will now talk about.
If you cannot leave your king vulnerable to capture it also follows that you cannot move your king into immediate capture, rendering them illegal too. There can be situations where no legal moves can be played but is not in check and so has not lost. This is called stalemate and in variant discussions is still a legal quagmire. In orthodox chess it's considered a draw, but this has not always been the case. Stalemate used to be a loss on either end: you lose because you missed out on a chance to mate or he loses because king capture is inevitable nonetheless. The latter shows up in variants that do not invoke check and instead puts capturing the king as the goal.
Is it check?
Lawyer up, boys; there's more.
Consider the pin. If a piece is between its king and an enemy rider it cannot move out of the way as it leaves the king to capture but it can still give check. Pin Chess is motivated by the consideration that a pinned piece can't give check on the basis of such a capturing move being illegal. But the other king goes down first anyway, you'd argue, and this may be true, but this caveat over pinned pieces can only come from questions regarding check; if checkmate weren't an issue the other king will go down first and that will be the end of it.
This brings up something: Can a piece that can potentially move to capture the king but cannot actually check? The common view is that they do, citing the above instance of priority or the like. But you can also argue that they don't by citing the part about not being able to move anyway.
An example is pawn promotion. Some variants only allow promotion to a piece that had been captured before, most of the time a pawn that's stuck near the edge of glory waiting for someone to take its spot can check, though it can be argued that if a pawn can't promote it's forbidden for it to cross to the edge of the board and get its upgrade and therefore cannot just go over and sic himself onto the king. This situation only pops up in variants with a finite pool of pieces to promote to, in practice captured pieces.
These only come up in situations that ask "What are the ramifications of not allowing this specific move on the basis that it leads to self-check?" The more common question is the one before it: "Let's mess with checking, why not?"
So while casting and en passant may be rules that variant makers get fussy over, it's nothing compared to the legal quagmire that is check rules.
Anyone can be king
The king's characteristics as a piece are so intertwined to each other that any piece that moves the same way in a variant risks the assumption that it is also royal. "royal piece" and "piece that moves like a king (Betza FW)" are independent qualities, to state the obvious, so we'll fiddle a bit with the latter and see what we can find.
The results aren't much in most cases. In cases of leapers and/or short hoppers the piece stops where it ought to and you simply get a king that moves funny. A change in leap movement does mean different mating ideas, but this doesn't make the target any more slippery; common usage for such pieces is to demonstrate mostly geometric themes. In practical variants there's rarely a need to improve on the king, who is nimble enough on his own.
Speaking of nimble, what of giving royalty to the sliding guys? A royal rider does seem to be a harder one to catch since it will go far away. The common answer is by restricting them from riding past check, i.e. a rider cannot only land on a square where it would be in check but also cannot move past it.
The fact that most variants never bother changing how a king moves but still manages to make playable games probably has got to do with the consequences of having "capture this piece to win" be the main goal to a game.
Pretenders
Royalty is the property of a piece where its capture means a loss, this is the first link in the chain that leads to check rules and the like.
We can change the part about capture to be any other condition a piece has to be in to merit a loss and the substance of royalty still stands, i.e. losing the game by its capture is accidental.
One flip is making checkmate the goal, the premise of selfmate problems. Even as a problem it's hard to play, thus the creation of reflexmate. Orthodox rules on checking still stand, just that what's usually the loss condition wins.
Another way to invert royalty is to make it so that it's in check when not threatened by capture, therefore checkmating it by depriving it of being targeted, a piece you rein in by giving more room.
Parton created another way: a piece that you cannot check but you can put into check which must then be removed by your opponent. If you cannot avoid checking you lose. Since it's easier not to check than check winning is mostly through forces.
Too many kings
The base material of this section is G. P. Jelliss' notes on royal pieces on Variant Chess magazine(VC 4, pp. 37-39). In the last part he mentions situations of multiple royal pieces and that "this is a subject for a future article." The article never did come on the magazine itself, but he has defined it on his glossary.
In such multirex situations a practical approach is to allow capturing royal pieces up to the last man where check stipulations kick in. This gets rid of some of the headaches that may arise from practical play, but as the next paragraph shows, these legalities make for great stipulations.
For chess problems, Jelliss gives three general stipulations. The simplest is a groupmate: The rule that you cannot leave a royal piece in check stands even with multiple of them, so a fork or skewer on two of them is checkmate enough. Groupmate-type loss conditions are also used in practical play.
Supermate is groupmate with the stricter requirement that all royal pieces be checked. If a move checks some but not all the royals is it illegal? The questions asks what happens after this check: If there is no way to resolve the checks then the receiving end has no legal moves left but is not in a loss condition and is considered stalemate. Checking in this way may also be made illegal, but in my opinion is an ad hoc rule.
Monomate only needs you to checkmate one royal, a loose but open-ended idea: how expendable are the others in this case? Must you only exact mate on one royal? Can you check the others? Can another royal stop mate?
Some games worth mentioning using this multiple royalty madness: Kinglet chess and Extinction chess. Both games require that the every single royal piece be captured, in Kinglet it's pawns while in Extinction at least one piece type must go(e.g. king, queen, both rooks).
Pawns are considered royalty in Kinglet that even promoting a pawn counts as losing it, the question of "are promoted pawns still pawns" only matters a lot in games with drops.
Extinction chess is interesting in that technically every piece is royal, but not equally. On the board are six different multirexes that mingle with one another, but operate in groupmate terms wherever applicable, e.g. forking two rooks is fine but a lone rook is a liability.
But these are nothing compared to All-mate Chess, where capture is through checkmate. To clarify, if a piece is attacked and cannot escaped it through regular chess means, it gets captured in All-mate. This is more a capture variant than a real multirex as king capture is still the goal and multiple captures are settled one at a time.
