The Constitution of The United States allows for the states as well as Congress to propose amendments to the Constitution. (Article Five of the Constitution.)
Whenever (including recently) anyone has suggested that there be a Convention of the States to propose new amendments, critics have raised the hobgoblin of a "runaway convention" and then they refer to the Convention of States as a constitutional convention, thereby muddying the waters.
Part of the problem is the myth that the Constitutional Convention of 1787 was a runaway convention. Well, there never was a runaway Constitutional Convention at any time in American history. Tons of historians are wrong about this!
Under the Articles of Confederation, which was the original constitution of the United States for the first decade of its existence, the 1787 Convention was not unconstitutional because the states withheld their sovereignty under the Articles, making Congress a mere creature of the states. The federal government was not the boss of the states, and if the states wanted to give their delegates in a convention a mandate to write a new constitution, they could and most did.
Nine out of 12 states at the Constitutional Convention (yes, there were only 12 - Rhode Island sat it out) had full mandates to write a new constitution; 1 state had a partial mandate; 2 states had no mandate. This is why most of the Massachusetts and New York delegations walked out of the Convention - they had no mandate to write a new constitution.
Delaware had a partial mandate. Technically, they should not have signed on to Article One because it created a Congress with several voting representatives from each state whereas Delaware's government had told its delegates to the Convention that each state should only have one vote; but everything else in the Constitution was kosher as far as Delaware was concerned.
So while the 1787 Convention itself was not a runaway, there were some half dozen runaway delegates (out of a total of 39 delegates). Alexander Hamilton was chief among them. The rest of his delegation walked out because New York had not authorized its delegation to write a new constitution. Hamilton went completely renegade when he actually signed the Constitution. Massachusetts also was not supposed to write a new constitution, but one of their delegates, Elbridge Gerry (the pol for whom "gerrymandering" was named) struck a conscientious compromise: he participated in the debate over the Constitution, but he declined to sign it.
Originally, Article Five only allowed for amendments to be proposed by Congress, but George Mason, a delegate from Virginia, insisted that the article should include language allowing the states to propose amendments. This language was added at the Convention.
The Constitutional Convention of 1787 thus made it unconstitutional for there to be another constitutional convention, because, in ratifying the Constitution, the states each gave up a portion of the autonomy they had retained under the Articles of Confederation, which autonomy had allowed them to shoulder aside the existing constitution, and the Congress created by it, and write a new constitution. The new Constitution was stronger than the Articles and gave the new national government power over the states that the old government never had.
Along with erroneously conflating a Convention of States (which is constitutional) with a Constitutional Convention (which is unconstitutional), critics who invoke a chimerical concern about a "runaway convention" often complain that a convention might change our Constitution in ways that undermine the rights of both the states and of individual citizens.
This worry is mistaken on two counts. One, a Convention of States can only propose amendments and, to take effect, these must be ratified by a super-majority of the state (three-fourths). Two, our Constitution has already been rewritten and is being rewritten with and without benefit of lawful procedure. A Convention of States is a legal way for the states to determine the limits of the federal government for a change, rather than having the federal government decide its own limits.
What might be in the amendments proposed by a Convention of States? A number have already been proposed. One popular suggestion has been term limits for members of Congress. Interestingly, Sen. Ted Cruz has just proposed a term limits amendment for Congress to consider. If it is successful, it would be unnecessary for a Convention of States to consider the same proposal; however, this is not the first time that such an amendment has been proposed, and it has failed in the past. Senate Majority Leader Mitch McConnell is adamantly opposed to such an amendment.
A Convention of States is not a Constitutional Convention and cannot change the whole system of government. It can only propose specific reforms, the more precise the more likely to be ratified by the rest of the states. Thomas L. Pangle, professor of government at U. of Tex. (Austin) has suggested that perhaps there is no middle ground between a powerful national government and a confederation of autonomous states ready to break apart and go it alone, but the Convention of States could be the mechanism by which the states can assert their power to counterbalance that of the federal government by passing and ratifying amendments that will force the federal government to do what the states want, in place of what we have now, which is a system of the states gradually (or rapidly) losing more and more of their autonomy to an expanding leviathan of a national government.