I'll start by giving some on the fly:
Movies:
Se7en(1995), Thriller - A well-made and elaborate movie about two detectives hunting a serial killer.
Fight Club(1999), Drama - A story of two men forming and underground fight club. Can't say much without spoilers.
The Prestige(2006), Drama, Mystery - Two magicians rival each other in attempt to make the ultimate stage illusion.
Three Kings(1999), Action, Comedy - A satire about the Persian Gulf War.
Saving Private Ryan(1998), Action, Drama, War - After the landing at Normandy a group of soldiers goes on a mission behind enemy lines to save a soldier.(Cried like a newborn baby.)
The X-Men movie series - because X-Men. and explosions.
Lord of the Rings Trilogy(2001-2003), Adventure, Fantasy - waaaay too much stuff happens here for me to sum up. Basically epic quest, fantasy world, forces of evil type-stuff.
Saw(2004-2010), Horror, Thriller - A series of movies following Jigsaw, a murderer that likes to "play games" with people that usually lead to their death or a lot of pain and suffering.(only if you like gory horror movies.)(The early ones were better in my opinion.)
TV Shows:
Game of Thrones - Based on George Martin's series of books set in a fantasy world, complete with blood, sex and dragons(!).
(I don't watch much TV)
Anime:
Death Note(2003-2006), Psychological Thriller, Supernatural - A student finds the book of a god of death, which grants him the ability to kill people simply by writing down their name in the book.(I liked the first season more than the second one)
Code Geass: Lelouch of the Rebellion(2006-2007), Alternate history, Drama, Mecha - A former prince receives a power called Geass that using it he strives to obliterate the supreme monarch legion of the empire Britannia.
Steins;Gate(2011), Science Fiction, Mystery, Romance - A scientist accidentally invents a device that allows to send text messages to the past.
Also, I am hesitant to recommend No Game No Life until they actually make that second season :/