There's a Persona 5 anime coming. Atlus announced it earlier today. It's called Persona 5 the Animation and there's already a short trailer.
— モルガナ_ペルソナ広報 (@p_kouhou) July 30, 2017
Atlus appears to be producing the show with A-1 Pictures and plans for it to begin airing in 2018, two years after the game was originally released in Japan. Here's the teaser via Anime New Network (the YouTube video is currently region locked):
There's no news yet on any plans to localise the show for Western audiences. Persona 5 The Animation: The Day Breakers was released prior to the game in Japan last fall as a single episode prologue. It's currently available to stream on Crunchyroll and you can read our review here.