Beatrix/Chronology

Youth
Beatrix comes from Elfland. From an early age, she began to practice with her sword, and continued for many years, but as time passed, she began to feel uncomfortable in social settings due to her devotion to the sword, which she then left her country to concentrate on her commitment to the sword. In her youth, she earned the title "Goddess of War" after her victories in battle and winning the Bushin Festival tournament 10 years ago. Her sister was married and had a daughter, while Beatrix vowed to travel the world in search of worthy opponents to test her mettle with the sword.

Starting a Journey
Shortly after the death of her sister, Beatriz, the renowned Beatrix began a journey searching for signs of life of her niece, (presumably Alpha), apparently being the only living relative she has left. Due to her travels, Beatrix likely didn't learn from Beatriz that Alpha contracted Demonic Possession from a very early age, and was exiled from the elven country and her family.

Bushin Festival Tournament
Having heard rumors of a female elf that resembled her, Beatrix travelled to the Midgar Kingdom, hoping this mysterious elf was her long lost niece. Upon her arrival, she headed to the Bushin tournament where she met Cid Kagenou, and found the scent of elf on the boy. Interrogating him, she asked if he knew an elf who shared a resemblance with her. Met with a disappointing answer, she then tested Cid by swinging her blade at him, stopping right before his neck. Contrary to her expectations, the boy fell back to the ground and offered a generic scared reaction. Apologizing for her misjudgment, the two parted ways, but not before she complimented Cid, declaring that he will surely become a strong Dark Knight.

The following day, Beatrix went to the tournament's reserved seating and gained entrance due to her invitation with Iris Midgar after she won her match yesterday. Once again, she asked the spectators if they had seen an elf who resembled her, but to no avail. The spectators then asked her if anyone has caught her eye, attempting to validate her title of "War Goddess", and she replied that Cid had caught her interest, making the spectators skeptical of her abilities considering his displayed prowess.

Beatrix witnessed Iris Midgar being easily defeated by the mysterious duelist, Mundane Mann, and was impressed with his strength, desiring to face him. She watched as Mundane Mann them revealed himself to be Shadow, so Beatrix stepped in to face Shadow and the two engaged in an intense battle.

Novel/Manga Outcome
But it was clear Beatrix was losing, as Shadow's technique was on a whole other level. Iris Midgar decided to help Beatrix defeat Shadow, but they were easily defeated by him.

Beatrix interrogated Shadow that, if he has so much power, why he doesn't he finish them off. Beatrix was perplexed as he replied that he was not interested in taking their lives, that he only destroys his "true enemies." Before he disappeared, Shadow reminded her and Iris that they should not lose sight of the real enemy, before he escaped in a flash of energy, leaving behind everyone surprised in the coliseum.

Anime Outcome
During her initial fight against Shadow, seeing that he had overwhelming speed, capable of moving to such a degree that she was not able to perceive him, and that she could barely react to joust, (although this was only because Shadow allowed it), Iris Midgar proposed to help Beatrix to defeat Shadow, using a Sword of the Three Heroes, and their fight spread to the corners of the capital of the kingdom of Midgar. In their fight Beatrix asks Shadow if they had met before, to which he replies that who knows, since he never forgets powerful opponents, to which he asks her and Iris, if they were deserving to be remembered by him.

Their battle continues until they reach a tall clocktower, where seeing that they were not achieving anything against Shadow, Iris proposes to immobilize him even if it costs her her life, so that Beatrix can then attack him, but Shadow found out about her move and easily subdued them against their ground by giving them a crushing gravity kick (contained), which made them suffer a lot, leaving both Beatrix and Iris with little strength to fight. When Shadow was about to leave, Iris reproached him that now the entire kingdom of Midgar, and the kingdoms of the world have him as an enemy and that he will have nowhere to escape, to which Shadow laughs and responds strongly, questioning the need for him to run away. He then deploys his most powerful technique throughout the capital of Midgar.

Beatrix and Iris Midgar layed on the ground helplessly, to which their only option was to witness what Shadow was going to do. When tension was at its highest, everything was revealed to be but a trick by Shadow, who disappears without further ado, clearing the rain and lightning and leaving the kingdom of Midgar the light in his stead.

The next day Beatrix says goodbye to Iris, but not without first advising her to try to refrain from obsessing over this loss, since there are many entities that mortals cannot yet rival, and that she is still very young, to which Iris replies, telling Beatrix not to worry, and that she can't stay still for too long. While she was leaving on the train, Beatrix felt a little sad when she couldn't find any clues about her niece, but at the same time, she felt happy for having met people who came to her like Cid Kagenou.