Death Note is a Japanese animation series (or  anime for short)  written by Tsugumi Ohba. The main themes in this  series are 'psychological' and 'supernatural'. Recommended age rating is  teenagers and up.
Plot Overview: The story  starts with a teenage boy named Light Yagami, sitting in school and  obviously bored when he notices a note book outside his class and  falling from above. He later picks up the note book out of curiosity and  sees the words "Death Note" written on the cover with an odd set of  directions inside on how to use this note book to kill with simply a  name an face. He takes it as a joke but decides to keep it and give it a  try just for the hell of it. To lights amazement the Death Note is real  and Yagami Light suddenly has the power to kill anyone he desires with  only their name and face.
Light is then visited by the owner of the Death Note,  A Shinigami named Ryuk. Ryuk decides to fallow Light around because  he's bored of his own world and decides it might be fun if a human  possessed the powers of the Death Note. Ryuk fallows Light on his quest  to cleanse the world of all evil but never interfearing. 
Light  decides to use this power for justice, a no tolerance judgment for all  criminals and becomes the the persona "Kira". The governments start  taking notice of the odd chain of death with all inmates and hire the  famous detective "L" to track down the killer. It soon becomes a battle  of wits, traps, ploys and deceptions as both Kira and L seek to learn  the others identity. Kira and L are both wrapped in cloaks of mystery  while Kira can't kill without a name and L can't arrest without a name.
Personal Opinion: Death Note is really a jewel of an Anime   series but the ones who don't like it usually tend to hate. The story  is well thought out for the most part. although there are a few episodes  when you can't help but feel its going no where fast. The artwork is  also above average but the soundtrack isn't anything special. Although  it does have a very nice opening videos and music. Speaking of audio, do  yourself a favor and watch the original Japanese version with subtitles  if it doesn't bother you because the English Dubs on this series is  awful. The characters are  rich and believable with their original  voices while the English Dub voice actors didn't quit fit and the  emotion they acted out didn't seem as believable as the original  Japanese voice actors.
All in all, if you're looking for an Anime with a strong story, unexpected twists and and thrills then check out Death Note.
I give death note a 9/10
Related Information: There has been 2 live action movies based on the Death Note series and are in Japanese audio only.
There  is also a book, a novel written about cases L worked on before the Kira  case that took place in the story in the anime and manga series. The  novels title is 'Death Note: Another Note - The Los Angeles BB Murder  Cases'.
There is also a ton of merchandise available on this Anime, from Anime shirts and posters to actual Death Note note books. (Sorry they can't actually kill with a name but they can give a mean paper-cut.)