I don't believe that 5e has adopted the Gunslinger as a class. It might go that way, but I (for one) hope not.
Pathfinder had a rake of firearms-related classes and they never really sat properly with me. I'm happy enough with an early arquebus making an appearance (and an appropriate handful of Feats to accompany it), but PF was definitely straying into the Wild West Gunfighter genre with some of their later additions.
