Its power is only limited by the political constraints of Republican Party leaders worried about the long-term effects of Donald Trump winning or losing the nomination.
"He who writes the rules, rules -- as the old saying goes," said Gary Emineth, former chairman of the North Dakota Republican Party and expert on the party's rules. "I think it's going to boil down to how strong Donald Trump and Ted Cruz are and how close they are."
Who's on the committee?
The Rules Committee is comprised of 112 delegates --- mostly party leaders, longtime members of the Republican National Committee and lawyers who understand legislative process.
The 56 delegations to the national convention -- one for every state, the five U.S. territories and Washington, D.C. -- will each select two members for the committee each, one man and one woman. (Delegations will also pick their members for the three other committees: platform, credentials and permanent organization.)
Each one of the delegations will have significant power on its own -- if delegates supporting Texas Sen. Ted Cruz can win a majority on the Texas delegation, for example, they could pack the rules panels with two Cruz supporters.
The Republican National Committee will retain quite some power over the panel. Republican Party Chairman Reince Priebus will select the chairman of the panel from the 112 delegates selected and, like any committee or organization, that chairman will have significant power to control debate and the agenda.
Former New Hampshire Gov. John Sununu, long a leading voice among the establishment, was chairman in 2012.
How does it work?
The group typically meets a few days before the national convention to decide how the convention will operate -- to literally set the rules of the game.
In most other election cycles, the group simply takes the recommendation from the party's own rules committee (many of the same people serve with both groups) and approves it with little discussion.
But this year it is likely to face a heated battle before the convention even begins. Any package of rules needs to be assessed by all 2,472 national delegates and presented to the entire convention. Whatever package of rules wins the support of a majority of delegates present at the time the vote is called will govern for the 2016 convention.
Should the rules package be rejected, it goes back to the committee to craft another set of rules. The convention essentially cannot start until it approves new rules, exclusively for 2016.
"The first thing the convention has to do is establish the rules for itself. So each convention is a brand new, organic body that assembles itself, creates its own rules for procedure and various other things," Haugland said. "And once those are adopted, once the convention is in session, it is the highest authority of the Republican Party."
The majority of the committee will send its proposed set of rules to the convention, but if at least 25% of the members (28 delegates) can unify in opposition, they will have the chance to fight those rules on the convention floor before they're adopted.
If no nominee is selected under the approved rules after multiple ballots, the rules committee could reconvene midway through the convention and start the process over again.
What is Rule 40b?
Four years ago, when Ron Paul supporters stormed the Republican Party, Mitt Romney's campaign counsel, Ben Ginsberg, successfully pushed convention rules so that only a candidate who won the majority of delegates in eight states or territories could be nominated at the convention. Known as Rule 40b, it effectively kept Paul off the ballot at the 2012 convention -- and ticked off a lot of tea partyers and conservative activists.
This year, the Republican establishment has the opposite problem; Rule 40b could keep an establishment white knight -- like Mitt Romney or Paul Ryan -- from riding in to the rescue. Under that rule only Donald Trump and Ted Cruz would make the cut — and only if that rule is adopted in July.
In most conventions, the rules from the old convention are simply approved for the newest convention. But this time almost everything is a fight, and there's no guarantee what will happen. The Cruz and Trump campaigns are pushing to keep the rule, but Kasich delegates and party regulars could easily keep it out of the 2016 rules.
The vast majority of the delegates going to Cleveland are bound by the voting in their states to cast a ballot for a certain candidate for at least one ballot. On the second ballot, about 43% of delegates become free agents and if no one gets to 1,237 votes by the third ballot, about eight in 10 of those people are free agents.
But if the panel carries over the 2012 rules, there would still only be two men available for the delegates to choose from.
So what will happen?
The Trump and Cruz campaigns have predicted that the rules panel will be stacked with their supporters, giving them enough votes to box out Kasich.
"If there's a contested convention, 80% of the delegates are gonna be Cruz delegates or Trump delegates," Cruz told reporters in Wisconsin last week. "Both Donald and I have been very clear, we shouldn't be changing the rules because Washington is unhappy with how the people are voting."
Louis Pope, vice chairman of the Party Rules Committee and a Maryland RNC member, explains that the competing campaigns could push for just about anything if they win a majority on the convention rules committee and the majority of delegates.
But that is a tall order with three candidates in the race and all eyes on Cleveland.
"These things will be watched like a hawk," he said, noting the committee proceedings have been broadcast before on C-SPAN. "There's not going to be any shenanigans."
Marco Rubio's former state director in Louisiana, Lionel Rainey III, noted that although there will be supporters of Kasich and of other candidates who have dropped out at the convention, the external pressure will be felt inside the convention hall.
"I think the last thing that the party wants and the last thing that these delegates want is to turn this thing into an open market and have it be someone other than who voters have chosen," Rainey said. "These voters are politically astute. They've been involved, they watch the news, and they've seen the complaints that the convention is going to be stolen."
Trump himself put it a little more bluntly on CNN in March: "I don't think that you can say that we don't get it automatically, I think you'd have riots."