Pre-Series
Jasper was one of the Gems made during the Gem War as a response to the growing rebellion being brought on by Rose Quartz. She in particular came out of the Beta Kindergarten, as the perfect example of a Quartz Gem. During the war she was put under Hessonite's platoon. Like other Gems, she had been placed into the battlefield, and managed to survive throughout the entire war, evacuating Earth just before the Corruption Light took place. She was praised as a war veteran to Homeworld, and became well known throughout Gem kind.
Season 1
After getting a distress signal from Peridot (though turning out to be Sombra), Jasper arrived down to Earth to confront the local Crystal Gems, at first considering them a waste of her time. Her mind quickly changed however when she saw Steven Universe, who she thought was Rose Quartz. This intrigued her enough to ditch the original mission in order to show Yellow Diamond her, and captured her and the others, placing them into her ship. She informed Yellow Diamond about the situation before she was found by Steven, and they both got into a fight as her ship crashed onto the planet. Jasper survived, but was found by Sombra sometime later, and ended up captured.
Season 2
Preparation Arc
Having no method to return to Homeworld, she and Peridot had to stay with the Crystal Gems from there, Jasper showing the most distane for having to do so, seemingly breaking the door to the Beach House multiple times, and spending her days training down the beach. She was brought on a mission to the Prime Kindergarten, her showing little interest in listening to Garnet throughout the endeavour and calling her a embarassment. She was however confronted by a Cluster Gem, and during the fight, accidentally fused into Carnelian. She immediately jot the fusion up as a forced fusion, and she immediately stormed off.
During another training session, she was then found by the Armadillo Monster, and during the fight a portal opened up back into Equus, which she was knocked into. She found herself in Ramicorn. She ran into Cloven Hoof during her travels, but when warned about the path ahead, she basically ignored him, only to get caught up in a deadly storm. She considered it just a physical enemy, but finding she can't even harm it made her run back to the barn, almost getting shattered in the process. She traveled through the mountainous region with Cloven Hoof when the storm passed, and told the concept of a "conscience". After passing through, she was found by Discord and brought back to Earth, where she confronted Garnet about it. She calmed down and began to come to better terms after understanding the situation when the Jet Monster arrived. She was unable to fight it herself, so she and Garnet formed Carnelian again, and managed to pull it off. Course she got upset all over again when finding out about Garnet's future vision.
Homeworld Arrival Arc
During the ride back up to Homeworld, Amethyst found her and was asked about how Homeworld worked. Jasper gave her a basic sum up, and told her to follow the Diamonds' instructions, and basically try to fit in with her own cut.
Upon arrival, she, Ruby, Amethyst and Steven went to the training grounds when they first came across Rupee. She felt insulted with her in her insults, and this resulted in an oncoming duel for both of them to face. Beforehand, Jasper brought Peridot back to Yellow Diamond as signs of her successful mission, but was questioned a bit about the Crystal Gems and the Earth as a whole. After reporting on Kyra, they were allowed to go. When the duel started to come around, Jasper was given some advice from Garnet and Pearl, and the duel then commence. She at first was having an advantage, but started to get pummeled during the fight. She was only saved from shattering when Kyra jumped in and intervened in the fight, stopping the duel altogether. Unfortunately, this also got her banished from Yellow Diamond City as Yellow Diamond felt she lied to her. She did manage to regain some respect though during the Osicone fight, as she protected her diamond despite the banishment. In the end, she decided to stay with the Crystal Gems.
My Little Universe: Mindwarp
Jasper tried to wake up Steven once he was found sleeping through a bit more brutal means, only to find that it didn't work, making her question if Humans have a dullance to pain when they sleep. When told no, she and the others brought him to Equus, learning of Lucid, and going into the dreamscape in order to find him. She had little tolerance for Lucid's games throughout the whole thing, keeping the group focused on finding Steven, and when it came to her turn for torment, she flat out lashed out at Lucid and demanded for their game to end, making Lucid scared and no choice but to comply.
Season 3
During the wedding planned for Matilda Doodle and Cranky Doodle, she and Garnet went out of town in order to fight off a Bugbear for them, making sure that the wedding didn't get innterupted. She was also one of the Gems affected by Discord's Body Puzzler, her getting Peridot's body as hers was taken by Emerald. She kept an eye on him during this, as annoyed as she was. During the situation with Time Jumper, much like the other Gems, she ended up poofed and didn't come back until later on in the timeline.
Season 4
Jasper didn't discover Stevonnie until later on during her training, finding it fairly surprising upon initial discovery. Not long afterwards, she went to Garnet and Flint to figure out the situation, just to find the two arguing about the exact same thing. She simply gave the suggestion for Flint to watch them if it bothered him so much, sending him on his way so she could talk to Garnet.
When it came time for Steven's first solo-mission, Jasper gave her support to the idea, thinking that Steven should start getting mission like the Quartz he was. With extensive training however, she and Pearl were sent off to Las Pegasus to get some relaxation. She saw a movie seemingly about their adventure with the Osicone, which she didn't enjoy. That night, Jasper saw the dream Pearl had of Rose Quartz, but when confronting Pearl about it she acted too forceful, and ended up arguing with her until she herself walked away. Eventually though, she stopped herself and found Pearl again, apologizing for her behavior but making it clear that she had to move on. This act earned her her Cutie Mark, making her the first Gem to get one ever.
When Bismuth first arrived, she was told to keep away for a while in order for bismuth to get more suited before they reveal the truth to her, which Jasper does accept despite some varying questions in her mind.
When it was found that Sapphire had been taken by White Diamond, she went with them as a bodyguard to Lapis Lazuli as the disguise tactic, given a camouflage band by Flint to hide her Cutie Mark upon arriving to the White Diamond Space Station. During the visit however, she was caught by White Diamond cohorts, and during the mighty brawl, and later the duel between Earthstone, she took it upon herself to stop Lapis Lazuli by fusing with her, forming Malachite. She kept in control just long enough so Star Quartz could open a window, and make Malachite fall down to Earth.
Season 5
Jasper herself never really appeared during this season, with the only exception of her passed out not too long after Earthstone split up Malachite, only to be taken away by Grootslang and remain as a gemstone.
Season 6
Jasper returned to the Crystal Gems, first shown fighting off a Slingtail, beating the animal alongside Bismuth before Rainbow Dash could arrive. Her involvement in particular was a little bit vague when it came to the Pink Diamond case, though she did reveal that the Homeworld Gems were told to evacuate the planet, instead of being driven out as Garnet had figured. She seemingly told her involvement to Flint of her War days, though did question later on when he started to question Pearl's involvement, asking if he thought she had killed Pink. Before they could get a straight answer, she and the other Gems were captured and brought back to the Space Station. she remained there until the humans broke her out along with the others.
Season 7
After she was rescued off of the Jungle Moon by Blue Diamond, she remained with the other Crystal Gems down on Earth while half of the team went back to the White Diamond Space Station.
After she found out about the truth of Pink Diamond, she ran off back into the Beta Kindergarten in order to take out her frustration on the place. By the time Fluorite's components had arrived there, she made it perfectly clear to them that she did not want to be bothered by anyone, still putting her own existence and title as the perfect quartz into question. She later confirmed that she will return on her own accord, once she got things sorted out.
Season 8
White Diamond Arc
Much of her involvement during this arc was fairly small, as she spent much of it out on Gem hunting missions alongside much of the others who remained on Earth, often the one assigning which Gems go on which missions. During this, she assigned Emerald on a mission to the White Tail Woods to handle the local Timberwolf population as a means to give Twilight a break from all the work given to her.
Grogar Arc
After Grogar took control of Homeworld, she was approached by Steven, him asking her if she can teach him how to fight. Jasper did agree to teach him, also bringing in Emerald, Rhodonite, the Rutile Twins, Amethyst, Fulgurite, Pinkie Pie, and Peridot as extra students in order for more of the Harmony Gems to be ready to face Grogar when the time comes. Throughout much of the arc she placed them under her guidelines of training, including facing three corrupted Gems at the same time.
My Little Universe II
Jasper became one of the Gems to become rejuvenated thanks to Spinel's Rejuvenator. After being reformed, she returned to her default behavior of an aggressive Quartz Gem, constantly starting fighting with the local quartzes in Little Homeworld, taking materials away from the workers for training, and believing that Emerald was her superior and referring him as "clarity". Much of her time during this state was spent away from the Harmony Gems training. During one of these fights (in this case, with a Slingtail), she discovered Rhodonite nearby, demanding her to get out of the way only for her to freeze up. She managed to fight off the Slingtail, but was disgusted by seeing the fusion unable to stand up for herself, calling her a mess for trying to do a Quartz's job. A bit later, she decided to go out on a "Monster hunt" mission, only to find Lavi, Yellow Pearl, and Button running away from Leviathan's pod. When asked to help, she remarked how they first couldn't fight for themselves and now they think they can give her an order, demanding they make up their mind. After Button explained the whole thing to her, telling her how they would never be scared again if they fight her, Jasper only remarked with them being dumb instead of daring. She quickly fought against the pod and Leviathan herself shortly after this, defeating her fairly quickly.
She was later encountered by Hessonite when she started another fight with the other quartzes. Not remembering her at first, she challenged her to a fight, only to be quickly and humiliatingly beaten by her, Hessonite "putting her in her place". When being reminded on how Hessonite technically owned her since she was Emerald's Quartz, her head suddenly began to hurt as to her memory, thus being driven away. Fulgurite and Emerald found her later on in the northern forests, discovering that she got rid of all organic lifeforms in her space, including the grass. Here she was re-discovered by Rupee and she quickly challenged her out of revenge, to which Jasper happily agreed to. After hours of fighting however, she and Rupee then agree to a contest of strength suggested by Emerald and Fulgurite, which mostly was fighting off numerous creatures from Equestria, including Dragons (including Garble). By that night, she and Rupee tied and the tie-breaker turned out to be the Ursa Major's cub, the Ursa Minor. She herself suggested to fight the Ursa Major as the ultimate tie breaker once it showed up, but she was quickly beaten by the giant animal. This, accompanied by Rupee's mockery towards both the other Gems and Ponies, was just enough for her to regain all of her memory and get her Cutie Mark back. After this, she tells off Rupee for her mocking the Ponies, asking Rupee "what's there to love about [her]?" before leaving her with her friends.
She joins the Harmony Gems in both the following court session to get Discord and White Diamond's magic back, and the final main battle between them and Grogar's Army. Just before the main battle in the Realm of Corners, she joins the original Crystal Gems and Elements of Harmony to take out the weapons located on Homeworld, but got caught by Grogar and forced into fighting him. She and the others tragically got shattered during the fight. She and the rest were revived and reformed afterwards within the Supreme Kindergarten and join to stop Grogar's army from leaving Homeworld.