Instead, and in spite of weeks of U.S.-led airstrikes, ISIS has gradually extended its reach. The extremist group is now either present or in control of a huge swath of countryside, forming a 180-degree arc around the Iraqi capital from due north around to the west, and all the way to the south.
Around this zone there have been skirmishes, and occasionally heavy fighting, with Iraqi security forces and Shiite militias battling ISIS.
American jets have carried out more than two dozen airstrikes in the area, mainly near Fallujah and the city of Ramadi, further to the west.
Inside Baghdad itself, there are ISIS sleeper cells that carry out almost daily bombings and assassinations.
An Iraqi officer told CBS News that the airstrikes are helping to clear an ISIS-free buffer zone around the city, where there are Iraqi boots on the ground. In fact, there are 60,000 men assigned to defend the capital, and CBS News correspondent David Martin reports that there are 12 teams of American advisers deployed with the Iraqi brigades. The estimate is that the Iraqi army will fight for the capital and there is no real concern that Baghdad is in imminent danger, Martin says.
Still, questions remain as to whether the Iraqi are disciplined enough to put up a sufficient fight if ISIS launches a major offensive.