The Single Action Shooting Society or SASS is the worldwide regulatory organization of Cowboy Action Shooting. Cowboy Action Shooting is a form of shooting competion based on accuracy in typical Old West situations, ie: mounted shooting, or a "sweep" shooting event (see the videos below). 

SASS is said to have over 75,000 members representing all 50 US states and 18 foreign countries. This worldwide organization sponsors regional, state,and national matches conducted by its more than 500 affiliated clubs. SASS creates and oversees the rules and procedures to ensure safety and consistency in the shooting events. One of the most significant points SASS members share, is a common interest for preserving the history of the Old West.

Competitive shooters in the SASS seem to prefer the Colt single action army revolver. This is a reproduction of the 1871 patented Colt pistol used by many of the famous and infamous names in the old west. Names like Hickok, Earp, and the Wild Bunch have relied on its accuracy and reliability. There are other revolvers used in competition, but they must adhere to the rule that it must operate as intended by the original pre 1900 design, and must be at least .32 caliber and not more than .45 caliber.

As for rifles it is stated that any lever or slide action design dating from 1860 to 1899 including replicas that operate in the same peramiters having not less than a .32 caliber and not more than .45 caliber may be used.

The shotguns used in the events may be side by side or single barrel. The model design must date between 1860 and 1899. The gauge of the shotgun must be 20,12, or 10. The only slide action shotgun allowed is the Model 1897 Winchester.

Then you need to carry the revolvers into competition. For that you will need gun holsters. The SASS states, "All handguns must be carried in a safe holster capable of retaining the firearm throughout a normal range of motion. Main match holsters must be located one on each side of the belly button and separated by at least the width of two fists at the belt. Holsters may not depart from the vertical by more than 30 degrees when worn."

You should go to the SASS web site and complete the registration process to become a member. They ask for you to create a cool old west alias, but choosing one may not be as easy as you think.