"The Ancient West," known during the 20th and 21st centuries as "the Wild West" or "the Old West," describes the period of 19th century Earth when the western United States of America was being settled. This period in Earth's history would be romanticized in Earth literature and popular culture for centuries, most frequently in stories known as Westerns.
In 2153, the Enterprise NX-01 discovered a Human and Skagaran settlement on a planet in the Delphic Expanse that resembled the towns of Earth's Ancient West. It was noted by Charles Tucker that the resemblance was accurate down to the spittoons. (ENT: "North Star")
In 2268, the Melkot trapped James T. Kirk and his landing party in an incomplete recreation of the American Ancient West town of Tombstone, Arizona, where Kirk and crew were to be executed in a reenactment of the infamous gunfight at OK Corral. (TOS: "Spectre of the Gun")
In 2369, Worf, his son, Alexander Rozhenko, and Deanna Troi participated in a holodeck adventure set during the Ancient West in the town of Deadwood, South Dakota. The program was designed by Alexander and Reginald Barclay. Troi, Worf and Alexander played this program when an experiment being conducted by Geordi La Forge and Data went awry and influenced the progam, giving the main characters Data's physical characteristics and removing the holodeck safeguards, making the 19th century firearms deadly. (TNG: "A Fistful of Datas")