Other ideas floated by candidates include opening up gun manufacturers to lawsuits for misuse of their products and blocking gun ownership for more misdemeanor offenses related to stalking and domestic abuse charges.
Many Democratic contenders also have backed "red flag laws," which would allow family members, co-workers and others to petition the courts to have police temporarily confiscate guns from individuals who they fear are a danger to themselves or to others.
These laws already exist in many states, including Florida, where Republican Gov. Rick Scott signed such legislation in response to the Parkland shooting, a case in which the suspect exhibited warning signs of violent behavior.
Democrats have also called for repealing a 2005 law that grants firearms manufacturers immunity from lawsuits over misuse of their products.
Trump's position on guns
President Donald Trump has generally hewed closely to gun rights activists and the National Rifle Association and called for arming educators in response to school shootings. He supports expanding concealed-carry laws, opposes banning assault weapons and high-capacity magazines, and his administration has threatened to veto bills to broaden background checks to more sales.
But after the Parkland shooting and later those in El Paso, Texas, and Dayton, Ohio, Trump did express interest in expanding background check laws; however, he backed off committing to major new legislation. Congress did add some incentives to keep the background check system better updated.
The Trump administration has also encouraged states to pass red-flag laws, but has not yet backed bipartisan legislation to encourage their adoption or create a national version.
The president also ordered his administration to look into a ban on "bump stocks," which can make it easier to fire semi-automatic weapons at faster rates and were employed by the 2017 Las Vegas shooter. and a new federal rule went into effect this year banning their use and sale. But some Democrats argue the ban should be done through legislation as well.
What critics say about gun control
Gun rights activists argue many of the Democratic proposals would violate the Second Amendment, which the Supreme Court determined in a 2008 decision to include an individual right to own firearms.
Many also argue assault weapons are unfairly singled out given that they're a popular rifle and, while employed in many high-profile mass shootings, are responsible for a much smaller percentage of gun deaths than other firearm types like handguns, most of which would not be covered under a ban.
Previous state and federal assault weapons bans have still allowed similarly powered semi-automatic rifles, and critics argue the differences between the new and old weapons — like removing a collapsible stock or pistol grip — were "cosmetic" and not likely to lessen the risk of violence.
Critics have been strongly opposed to the "mandatory buyback" assault weapons proposal, arguing it would turn millions of currently law-abiding gun owners into criminals overnight if they failed to get rid of their weapons. The NRA estimates there are more than 16 million firearms that are currently in circulation that would be defined as assault weapons. In addition, recent state and local restrictions on types of guns or magazines have been plagued by widespread noncompliance, suggesting it may be difficult to enforce.
Some Democratic leaders have been upset with the buyback discussion as well, arguing it undermines their efforts to pass other gun legislation after years reassuring gun owners they would not try to confiscate weapons.
And while expanded background checks are overwhelmingly popular in polls, critics such as the NRA argue that they and related proposals like gun licensing are an unnecessary and burdensome addition to existing laws and that criminals will evade the new requirements by buying guns illegally.
The NRA has also argued legislation granting immunity to firearms manufacturers, which they successfully lobbied to pass in 2005, is necessary to protect the industry from a deluge of lawsuits that could put companies out of business.