The supreme court has not ruled on the interpretation of the second amendment to the US constitution since 1939, which states:
"A well regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed."
Heller argues that the DC gun ban, which also requires that rifles or shotguns must be kept in homes disassembled or with their triggers locked, violates that amendment.
But city officials say the ban, introduced in 1976, is necessary to keep violence and murder rates down, and that the second amendment only protects gun rights for people associated with militias, not individuals.
Adrian Fenty, Washington DC's mayor, said earlier this month: "I'm confident in our case, and our continued ability to protect residents from gun violence."
However, Alan Gura, the leading lawyer for the plaintiff, questioned the impact on the crime-rate of the city's laws, saying they have "accomplished nothing except to prevent law-abiding citizens from exercising their constitutional right to keep and bear arms."
The case has also split the Bush administration.
The department of justice has backed the mayor's position, while Dick Cheney, the vice-president, has joined a group of US parliamentarians in urging the court to adopt a stronger stand in favour of gun rights.