There are 5 HMs (or Hidden Moves) in Pokémon X and Pokémon Y. These give outside of battle effects as well as being moves useable in battles. Although not necessary for the completion of the story, these moves will enable you to traverse the entirety of the Kalos Region and find many useful items that will aid you on your journey. Each HM requires a badge in order to use it outside of battle.
Found in Parfum Palace gardens in the center of southwestern hedge maze.
Obtained from Professor Sycamore at the Seaside Station in Coumarine City
From your Rival at the Route 12 entrance in Shalour City
From Grant on the bike ramp north of the Cyllage City Gym
From Shauna after defeating Shauna, Tierno, and Trevor on the bridge on Route 19