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History of firearms and rifling 
Early firearms had smooth barrels, and fired projectiles with no significant spin. These projectiles had to have stable shapes, such as finned arrows or spheres, to minimize tumbling during flight. However, spherical bullets do tend to rotate randomly during flight, and the Magnus effect means that even a relatively smooth sphere will curve when rotating on any axis not parallel to the direction of travel (see knuckleball for an example of intentional random tumbling.)
A rifled barrel, having spiral grooves or polygonal rifling, imparts a gyroscopic spin to the projectile that stabilizes it in flight and prevents it from tumbling. This does two things; first, it increases the accuracy of the projectile by eliminating the random drift due to the Magnus effect, and second, it allows a longer, heavier bullet to be fired from the same caliber barrel, increasing range and power (see external ballistics). In the eighteenth century, the standard infantry arm was the smoothbore musket; by the nineteenth century, rifled barrels became the norm, increasing the power and range of the infantry weapon significantly.
Modern smoothbores 
Some smoothbore firearms are still used.
Small arms 
A shotgun fires multiple, round shot; firing out of a rifled barrel would impart centrifugal forces that result a doughnut-shaped pattern of shot (with a high projectile density on the periphery, and a low projectile density in the interior). While this may be acceptable at close ranges (some spreader chokes are rifled to produce wide patterns at close range) this is not desirable at longer ranges, where a tight, consistent pattern is required to improve accuracy.
Another smoothbore weapon in use today is the 37-mm riot gun, that fires non-lethal munitions like rubber bullets and teargas at short range at groups of people, where a high degree of accuracy is not required.
The cannon made the transition from smoothbore firing cannonballs to rifled firing shells in the 19th century. However in more recent times tank guns have moved back to smoothbore to reliably penetrate the thick armor of modern armored vehicles. A very long, thin kinetic-energy projectile is required. The longer the projectile is in relation to its diameter, the higher the spin rate must be to provide stability. Practical rifling can only stabilize projectiles of a limited length-to-diameter ratio, and these modern rounds are simply too long. These rounds are instead formed into a dart shape, using fins for stabilization. With the fins for stability, rifling is no longer needed, and in fact the spin imparted by rifling would degrade the accuracy of a finned projectile. The first tank with a smoothbore gun was the Soviet T-62, introduced into service in 1961; today all main battle tanks except the British Challenger 2 and Indian Arjun MBT use smoothbore guns. The Russian navy conducted experiments with large-caliber smoothbore naval guns, which were halted by budget cuts.
The armour-piercing gun evolution has also shown up in small arms, particularly the now abandoned U.S. Advanced Combat Rifle (ACR) program. The ACR "rifles" used smoothbore barrels to fire single or multiple flechettes (tiny darts), rather than bullets, per pull of the trigger, to provide long range, flat trajectory, and armor-piercing abilities. Just like kinetic-energy tank rounds, flechettes are too long and thin to be stabilized by rifling and perform best from a smoothbore barrel. The ACR program was abandoned due to reliability problems and poor terminal ballistics.
Mortar barrels are typically muzzle-loading smoothbores. Since mortars fire bombs which are dropped down the barrel and must not be a tight fit, a smooth barrel is essential. The bombs are fin-stabilized.
See also 
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