The US Constitution’s 2nd Amendment says the right to keep and bear arms shall not be infringed. Mostly this debate is centered on firearms, but many other forms of military grade equipment are banned. Private citizens cannot own nuclear weapons, WMD’s, missiles, advanced fighter aircraft etc, and this is commonly accepted. How and when did the practice of presenting citizens from possessing advanced weaponry start?
On a national level, restrictions on "heavy" weapons were put in place by the National Firearms Act of 1934 in reaction to the gang violence sensationalized during the Prohibition Ere. Notably, the St. Valentine's Day Massacre is commonly cited as the impetus for the Act, as it was a highly publicized event committed with machineguns. NFA, as it's become known, didn't actually explicitly ban anything, but rather instituted a registry and a $200 "tax" on the items in question along with fees for transferring said items. The items impacted by the NFA are:
The short-weapon restrictions may seem nonsensical given that handguns aren't included in the NFA, but that was because handguns were pulled from early drafts in order to secure passage of the law, leaving for some unusual loopholes.
Nowadays, the $200 tax and associated ~6-month wait for applications to process are more of an inconvenience, but in 1934, the NFA effectively amounted to a ban on these classes of items. Oddly enough, when challenged, it made it all the way up to the Supreme Court in the 1939 case United States v. Miller. The case remains a bit contentious to this day due to the fact that the defendant (Miller) died before the case was heard and never actually presented any arguments. At the same time, the US attorneys included arguments that don't exactly follow modern ideas about "common sense" restrictions - a key piece of their argument centered around the idea that the Second Amendment only protected weapons considered suitable for use in a militia, and that a sawn-off double-barrel shotgun in question was not suitable for use in a militia and therefore was not subject to the Second Amendment,
The Supreme Court held a unanimous decision in favor of the government that the Second Amendment was not violated, with the majority decision largely following the argument that the items specified in the NFA were not protected by the Second Amendment:
In the absence of any evidence tending to show that possession or use of a "shotgun having a barrel of less than eighteen inches in length" at this time has some reasonable relationship to the preservation or efficiency of a well regulated militia, we cannot say that the Second Amendment guarantees the right to keep and bear such an instrument. Certainly it is not within judicial notice that this weapon is any part of the ordinary military equipment, or that its use could contribute to the common defense.
So, oddly enough, the very law that makes most heavy weapons unusually difficult to obtain (by American standards) was upheld by a Supreme Court decision that almost seems to make the argument that traditional weapons of war are protected by the Second Amendment.
That's really the basis of all of it. There's a patchwork of other laws that came together since that have further restricted things, but the biggest piece was the 1934 NFA.
US v. Miller decision for the quote: https://www.law.cornell.edu/supremecourt/text/307/174#ZO-307_US_174n2
To address one specific point: private ownership of nuclear weapons is regulated by the Atomic Energy Act of 1946, which puts heavy restrictions on many categories of nuclear technology. (Private ownership of nuclear reactors is also regulated — you can only have on with a license, otherwise it is illegal.) While nobody in 1946 thought a private citizen ought to own a nuclear weapon, the issue at the time was less about whether some very wealthy individual might build one (that seemed impossible at the time; it took the combined industrial might of an entire country to produce them then, though today you could imagine certain mega-corporations with the appropriate expertise being able to pull it off on their own), but rather about research and ownership rights. Not only can you not own a nuclear weapon in the US, you can't patent one, either — they are one of the strange exceptions in US patentability, in part because of fears of secrecy and fears of a lack of government control. (If you're curious for more, I wrote a very long article on this awhile back I know of only one instance in which the US threatened to enforce the "ownership" aspect of the law: in the late 1960s/early 1970s, a private company tried to develop a fusion reactor that had characteristics that were similar to how H-bombs work. One of the ways the US Atomic Energy Commission tried to persuade them to stop their work was to tell them that if they were successful, it would be classified as a privately-created nuclear weapon and they'd arrest them all. In the end, they couldn't get it to work, so the issue was moot (and a bit of a stretch anyway; I'm not sure it could have survived the legal challenge). But it's an interesting historical tidbit. (This is from as-of-yet unpublished research of mine.) I have never seen any serious debate where it has been argued that the Second Amendment should be construed as allowing private ownership of nuclear weapons (though Scarry, in her somewhat odd Thermonuclear Monarchy, argues that nuclear weapons conflict with the Second Amendment, though it is a very different sort of argument — she claims that this should lead to the banning of nuclear weapons, not their private ownership).