In the manga, Serenity was fascinated by the Earth. Even though she was given orders not to go to the planet, she secretly visited there and met Prince Endymion. The two then fell in love. During the Dark Kingdom's attack which caused the Moon Kingdom's downfall, Prince Endymion died protecting Serenity. She was in grief due to Endymion's death, and committed suicide. Serenity's mother, the Queen, sealed away the evil and the moon (as seen in Volume 3), but everyone involved was killed. Before her own death, the Queen used the Silver Crystal to give her daughter and others another chance at life, hoping in particular that, this time, Serenity and Endymion could find happiness together.
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In the anime, Prince Endymion would sometimes visit the Moon Kingdom to see Princess Serenity. One day, not long before a planned attack on the Moon Kingdom was about to take place, Endymion visited the Moon Kingdom to warn Princess Serenity. Guards saw him and tried to chase him away. At the ball, the Prince appeared to greet the Princess at the staircase in disguise wearing a tuxedo and mask. He told the Princess that danger was coming, but he lingered with her too long and got caught up in the battle.
When he tried to defend Princess Serenity from Queen Beryl, the two were separated by a gust of wind which pulled the Prince up towards Queen Beryl. When the Princess jumped to reach her love, Queen Metalia killed both of them with an energy blast.
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Okay, so? Serenity is the only child and heir of a powerful, ethereal queen (who rules the moon, no big deal). All of her life, she's been greatly sheltered, leading to her not knowing much of the universe outside her castle walls. Much like her reincarnated counterpart on which the Sailor Moon franchise focuses, Serenity is a sweet, albeit naive and sometimes childish young woman whose love for those around her is ultimately her greatest strength - and downfall. However, it seems that a degree of melancholy is present in her that isn't in Usagi/Sailor Moon; her story, after all, is ultimately a cosmic tragedy.
tl;dr she's a space princess. Slightly prim, a little proper, mostly emotional. Kindhearted but with an air of sadness. Otherworldly. Tends to love fast and destructively.
What I want to do with her in RP: Explore her differences from Usagi (are she and her reincarnation carbon copies in both looks and personality?), theorize how she views royalty and her future duties, develop her outside of canon scenarios (ie, put her in high fantasy or space opera settings), play with both her hinted-at ridiculous powers and her differences from baseline humans, and basically see if she could have ever become a badass, mature queen under all that sweetness and light.
Also, fluff. And smut. Lots of those.