2. Button Rifling
Button rifling is a modern method that creates the grooves in the cold surface of a rifle bore by displacing metal using a bullet-shaped, super-hard button of tungsten carbide. The rifling button has the reverse pattern of the groove profile ground into its surface. As the rifling button is pushed or pulled through the barrel, the groove pattern is ironed into the bore surface by displacement. There are several variations in button-rifling procedure. Some barrel makers prefer to pull the button through the bore, while others prefer to push it through. In most cases, the button remains free to rotate during this process, dependent on the angle of the grooves in its surface to cause the desired degree of rifling twist. As variations in rifling twist may occur during this procedure, some barrel makers affix the rifling button to a rod and positively rotate it with a sine bar.

Advantages of button rifling:
The procedure is fast and very economical, as only a single pass of the button is required to rifle a barrel.
Button rifling is well suited to mass-production methods with high output. 
Button rifling leaves a smooth, bright finish inside the barrel that need not be lapped.
Button-rifled barrels are very accurate.
Bore and groove dimensions are very consistent.

Disadvantages of button rifling:
Button rifling creates stress in a barrel; high-quality button-rifled barrels must be stress-relieved after rifling.
Buttons are expensive and difficult to make.
Different groove configurations and different rifling twists require a new button.
The button-rifling system is not flexible.

Button-rifled barrels can be extremely accurate; more bench-rest records are held by shooters using guns with button-rifled barrels than by any other type. Button-rifled barrels are very common on modern centerfire and rimfire guns.

To be continued

4. Hammer-Forged Rifling
Hammer forging is an ultra-modern method of rifling a gun barrel that is well suited to high-volume production by large manufacturers such as government arsenals and commercial corporations that can afford the sophisticated machinery. This method begins with a metal barrel blank about 12 inches long and 2 inches in diameter with a hole in its center honed to a fine finish. A tungsten carbide mandrel with the pattern of the rifling lands and grooves machined into its surface in reverse relief is then inserted into the hole of the blank. A forging machine with a series of radially opposed hammers is then used to compress the blank inward against the mandrel. As the hammers compress the outer surface, the blank is reduced in diameter and lengthened, simultaneously creating the bore and rifling. If needed, hammer forging can form the chamber and throat as well as a fully profiled outer surface. The spiral tracks of the hammers can often be seen on the outer surface of hammer-forged barrels. Some manufacturers turn the barrels to remove this surface, while others leave it in place.

Barrel blanks may be hammer forged cold or hot. Hot hammer forging reduces the amount of effort that the hammers must exert on the blank, and can result in better grain structure and improved strength. However, hot hammer forging is more expensive and requires more sophisticated machinery. The cold hammer-forging process produces barrels of excellent quality. Hammer-forged barrels are very common on high-volume centerfire hunting rifles and pistols where their consistency and strength outweigh their accuracy capabilities. They are not common on match-grade or varmint barrels, as their accuracy is perceived to be inferior to cut- or button-rifling methods.

Advantages of Hammer Forging:
Hammer forging consistently produces high-quality barrels. 
It can form chamber, throat and outer profile if necessary.
It does not remove metal; there's no waste or chips.
It produces barrels with excellent grain structure and high strength.
Hammer forging produces superb bore finish, no lapping needed.

Disadvantages of Hammer Forging:
Machinery and mandrels are expensive.
It's inflexible; changes in rifling require a new mandrel.
Quality is very good, but not match grade.
Hammer tracks are left on outer surface.
Process introduces stress in the barrel; must be stress-relieved.

To be continued.

Great thread, thank you texasmac.

Very good-answered lots of questions floating around in my mind!

you are a great source for information  thanks  art

"Cut rifling must be lapped."
This isn't true, as many of the best old barrel makers were adamant that they never lapped their barrels. By taking extremely small cuts each pass their rifling was perfect when finished and no need to lap their barrels.
Harry Pope was said to have gotten very angry at someone who simply asked if he lapped his barrels.

And I failed to include an even more modern process referred to as ECM rifling, which uses electrolysis.  Electrolytic machining (ECM) is a subtractive method operating on the principle of anodic metal dissolution via an external DC power source.

One of the critical features of ECM is there’s no contact between the tool (-) and the workpiece (+). The shape of the tool cathode determines the shape of the material removal. ECM is an imaging method. Rather than a cutter, broach or button, a rifling cathode is used.  Similar to the shape of a rifling button, the cathode has the reverse pattern of the groove profile with insulating material where no work is needed (lands), leaving the conductive material visible in areas where the material removal process occurs & forms the grooves. See photo below. The groove accuracy depends on the cathode design and the machining precision. This electrochemical process ensures creation of precise, polished grooves without any physical contact, applying zero stress to the barrel.

This EC rifling process is a departure from traditional processes as it delivers barrel rifling quality and productivity surpassing conventional capabilities. EC rifling is reported to be optimal for superior gun finishing, heightened productivity, and impeccable quality.

Wayne