Overall Story:
So you say to yourself, "Ok, Tenchi's on Cartoon Network but I'm too lazy to watch it.  I've been hearing awesome things about it but...well...I'll still haven't seen it." 
I then say to you, "What the heck is your problem??" and then apologize and do this for those of you who are pretty lost as to what's going on.  And ok guys, try and a be a little smart about this...

IT CONTAINS SPOILERS
A brief overview

Tenchi's just your average teenage, diligent, nice, friendly school-goer who happens to stumble upon a very angry, later beautiful demon that has it out for him.  Well, Tenchi kinda sets her free from the cave that she had been imprisoned in thousands of years before and she's not too happy about what she'd been doing for the past years. 

Enter Ryoko...
She terrorizes Tenchi around his school, trying to take out her energy on Tenchi.  She's a little perturbed.  Well, this involves the blowing up of Tenchi's school and him cutting off Ryoko's hand (which she regenerates with ease).  Later you see her asleep in Tenchi's bed, much to Tenchi's surprise!  This is how we meet Ryoko in the OAV series.  Seeing how I don't like the TV series all that much, we will leave it at that

Aeka, the "unfortunate" princess...
Later comes Aeka.  Ahh, that wonderful princess who's after Ryoko and later Tenchi.  She's looking for her brother, Yosho (and escaping her fiance, Seriyou).  She knows that Ryoko was the one that Yosho was chasing out into space because Ryoko kinda destroyed much of Jurai (where Aeka is from).  She tracks down Ryoko and from then on, they never get along.  There are many reasons that they don't like each other, but the main one is they both love Tenchi.

Sasami, the innocent bystander...
This is Aeka's super-cute, super-cook, sister.  Sasami is pretty mischevious and wants to play a "game" with Tenchi when they first meet.  This involves Tenchi getting that thing that Aeka wears on her head.  (It's really the key she uses to control Ryu-Oh, her ship, but that's a little more than you want to know if you're reading this).  Tenchi's less than successful, and one thing leads to another (crashing of ships into the Earth, blah blah, you know the drill) and they all end up living at Tenchi's house in the end.

Mihoshi, the bubbly blonde...
Due to her blondeness, this Galaxy Police Detective ends up crashing into the Earth and getting saved by Tenchi.  Stereo-typical blonde, she's everyone's friend and understands nothing.  What more can you say about her?

Then a whole bunch of other people come in and the story continues...

If you've never seen Tenchi Muyo! I highly recommend it.  This is your standard "boy is really confused by the girls from space that love him and this causes many problems, ie bad guys from outer space trying to kill everyone" story. 

If you liked El Hazard, Slayers or Saber Marionette J, Tenchi is pretty much a mix of that.  It's in a class by itself of course, but those are somewhat similar.
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