Current legislation in England and Wales is a "complex and confused" mess, and tighter restrictions on gun licences are needed, the Commons Home Affairs Select Committee warned.
Its review comes after taxi driver Derrick Bird killed 12 people in a shooting spree in west Cumbria in June.
He had been sentenced 20 years ago to six months in prison, suspended for a year, for stealing decorating materials. Bird also had a drink-driving conviction and had been arrested over claims he "demanded payment with menaces" after a customer made off from his cab without paying.
But the 52-year-old had been permitted to keep his shotgun certificate, which he got just before his 17th birthday, because neither of his convictions warranted an immediate custodial sentence to trigger an automatic ban.
The MPs said: "We do not believe it appropriate for those convicted of offences which are serious enough to warrant a custodial sentence to retain their firearms.
"We are also of the view that those who receive shorter custodial sentences should not be allowed to possess firearms and recommend accordingly."
The MPs findings come a month after the independent review ordered by Cumbria Constabulary proposed anyone handed a jail term of three months or more which is wholly suspended should be banned from owning licensed weapons for five years.
Committee chairman Keith Vaz said: "The terrible murders perpetrated by Derrick Bird in June highlighted gaps in the current licensing regime, notably around the ease with which convicted criminals can gain access to firearms.
"We have heard evidence of further cases in which individuals applying to obtain a licence for firearms have lied about their mental health problems and have subsequently gone on to misuse their weapons.
"Current gun law is a mess. It needs to be simplified, (and made) clear and consistent to be properly understood by both those using firearms for legitimate purposes and those in charge of enforcing the law."
Shooting organisations told the MPs that 34 pieces of legislation govern the use of firearms and few people could understand them.
The committee said this placed an "onerous burden" on police and members of the public who wanted to abide by the law and it called for one licensing system to cover all guns.
Minimum age limits should be considered "with the aim of reducing inconsistency and complexity around the use of firearms by children".
About 1,000 people under the age of 18 have a licence, including children as young as 10, who are not allowed to use the guns unsupervised until they are 15.
The MPs also called for the Government to consider requiring the police to consult the domestic partners of licence applicants.
Application fees should increase to cover the costs of gun licensing because of "concern about the potential impact of police spending cuts on the firearms licensing function", the MPs said.
But the committee said it was unconvinced by the idea of tagging the medical records of every licence holder and rejected a proposal to ban guns and ammunition being kept in homes.
The British Association for Shooting and Conservation (Basc) said it disagreed with the calls for minimum age limits, the increase in licence fees and applying the firearms licensing system on to shotguns".
Bill Harriman, Basc director of firearms, said: "This is not the end of the road in terms of the political battle to secure effective law which serves both public safety and firearms users."
Crime prevention minister James Brokenshire said: "Public protection is the first duty of any government and our firearms laws are among the toughest in the world.
"It is right that we keep them under review and we are prepared to tighten them further if necessary. Those controls must also be proportionate and fair."