Since the FBI started conducting the National Instant Criminal Background Check System (aka NICS), under the Brady Bill in late 1998, more than 220 million checks have been performed at the time of firearm sales:
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The simple reality, is that the gun background check system is working for the vast majority of firearm sales in the U.S.. Increasing checks for personal transfers is not going to take the many stolen guns off the streets that ultimately find themselves in the hands of criminals. Nor, will it stop those who may become dangerously mentally ill many years after a gun purchase. Nor, are increased background checks going to stop the near 90% of all violent crimes in this country that don't involve guns.
FBI: NICS Stats: https://www.fbi.gov/about-us/cjis/nics/reports/nics_firearm_checks_-_month_year.pdf
CNN: Polls: Gun ownership declines, support for less gun control on the rise: http://www.cnn.com/2015/06/19/politics/gun-ownership-declines-support-for-less-gun-control-on-the-rise/
Guns and Crime: http://www.gunfacts.info/gun-control-myths/crime-and-guns/
Property Crime has been reduced as handgun supplies increased: http://www.gunfacts.info/wp-content/uploads/2013/09/CRIME-AND-GUNS-Property-Crimes-and-Handgun-Supply1.png
Clinton's claim that 40 percent of guns are sold at gun shows: https://www.washingtonpost.com/news/fact-checker/wp/2015/10/16/clintons-claim-that-40-percent-of-guns-are-sold-at-gun-shows-and-over-the-internet/