What are the chances of encountering a shiny Pokémon?

For Sun & Moon, are the chances to catch a shiny Pokémon the same of previous generations, like X & Y? Or are the chances somewhat higher or lower this time around?

Breeding with the Masuda Method, your chances of getting a shiny are 1 in 683. If you have the shiny charm and use the Masuda Method, your chances are 1 in 512. Thus, using the Masuda Method is the best.

With SOS chaining, if you get a chain of 70 and above, you have a 1 in 1024 chance, and with the shiny charm it becomes 1 in 683. Therefore, the Masuda Method still gives you a better chance, but the SOS chaining method may be faster if you turn off battle actions in your game options.

Reportedly, they are the same as they were in gen6: 1 in 4096

The Masuda Method (breeding two Pokémon from different languages) still works to increase your chances of breeding a shiny. The way(s) of increasing the chances of wild shiny Pokémon have been replaced with Ally Chaining, which involves using an Adrenaline Orb to force the wild Pokémon to call an ally for help, KO one of the two, and let the remaining one call for help, until you encounter a shiny.