The two music stars have been arguing ever since falling out over some backing singers three years ago.
Their feud remains unresolved, with Taylor finally putting her music on Spotify on Friday - the very same day that Katy released her new album.
But this week, Katy said: "There are bigger fish to fry, and there are real problems in the world. You know what I'm saying? I love her, and I want the best for her."
The pair are hardly the first musicians to publicly lock horns, and in an age of Twitter there are more platforms than ever for the stars to trade barbs.
Here are just a few other high-profile musical battles.
Katy Perry isn't the only person to have found herself caught in a Swift storm.
Taylor's beef with Kanye West dates back to 2009 - specifically, that year's MTV Video Music Awards.
Kanye wasn't too impressed with Taylor winning video of the year for You Belong With Me.
And, let's be honest, Kanye is not the type of chap to suffer in silence.
The rapper took to the stage during Taylor's acceptance speech, grabbed the microphone, and said: "Taylor, I'm really happy for you, and I'ma let you finish, but Beyonce had one of the best videos of all time!"
He was talking about Bey's Single Ladies, which to be fair really was a masterpiece of filmmaking. But this perhaps wasn't quite the time or the place to start that debate.
The pair later made up, but a few years down the line West claimed in his single Famous that the run-in had helped Swift's career.
Taylor claimed she had never given West's lyrics her blessing, but then the rapper's wife Kim Kardashian got involved, posting a Snapchat which appeared to show Taylor doing just that.
At the 2016 Grammys, Swift responded: "I want to say to all the young women out there - there are going to be people along the way who are going to try to undercut your success or take credit for your accomplishments or your fame."
But, she advised her fans: "Just focus on the work and you don't let those people sidetrack you."
If a new female rapper arrives on the music scene and Lil' Kim (above left) doesn't start a fight with her - did she ever even exist?
Kim has history with Eve, Foxy Brown, Remy Ma and many more... but those were small fry compared to when Nicki Minaj appeared in the late noughties.
Nicki's debut album Pink Friday was a massive success when it was released in 2010, but Lil' Kim opted not to send flowers.
In a diss track she put out online called Black Friday, Kim described Nicki as a "Lil' Kim wannabe" who was "trying to clone my style".
Minaj, naturally, responded with a diss track of her own, called Stupid Hoe. Which drew attention to the fact that her records were "shipping platinum" while Kim remained "busted" and "crusty".
As a band, Oasis were no strangers to scrapes.
The infamous chart battle of 1995 saw Oasis's Roll With It go up against Blur's Country House in a race for the number one spot - a contest Blur eventually won.
The competition was so high-profile it even made headline news, much to the bemusement of a somewhat younger-looking John Humphrys.
But there was another Oasis feud was still to come - this time an internal one.
In 2009, the year Noel quit the band, he told Q his brother Liam was "rude, arrogant, intimidating, and lazy", adding: "He's the angriest man you'll ever meet. He's like a man with a fork in a world of soup."
Last year, Liam said Noel was absent from the premiere of a documentary film about the band because he was probably in one of his "really big houses, eating tofu and having a face peel".
Their brotherly battle rumbled on last week, when Liam criticised Noel for not appearing at One Love Manchester - a concert to benefit the victims of the suicide bombing in the city.
But Noel defended himself by saying he was never asked to play at the show, adding that he'd donated the royalties from Don't Look Back In Anger to the Red Cross's Manchester Emergency Fund.
Zayn Malik had everything a young man could wish for at the height of One Direction's success - fame, money and worldwide adoration.
But the man responsible for 20% of the group's success left Louis, Liam, Niall and Harry behind in 2015 to pursue a solo career.
Zayn began working on some new material with Naughty Boy - the producer famous for his work with Emeli Sande and Sam Smith.
But it's fair to say that he and Zayn didn't quite hit it off in the same way.
"Naughty Boy you fat joke stop pretending we're friends no one knows you," tweeted Zayn after their recording sessions turned sour.
Mr Boy later told The Guardian the 22-year-old was "throwing his toys out of the pram" and said they had fallen out over "external influences".
Hate it when that happens.
The number of artists which Azealia Banks (above left) has fallen out with is so huge that it's difficult to single out just one.
But somebody who experienced a particularly high level of vitriol was Fancy singer Iggy Azalea.
For two artists who became popular around the same time and were only a couple of letters off sharing a name, perhaps comparisons and rivalries were inevitable.
It all kicked off when XXL magazine put Iggy on the cover of their annual freshman issue, which looks at the hottest newcomers in music every year.
Banks was not a massive fan of this editorial decision.
She said the magazine was "wrong" to include Iggy and went on to accuse her nemesis of appropriating black culture.
Iggy Azalea responded by saying: "You can't block my blessings, today I'm celebrating, get on with it or kick rocks."
The battle rumbled on for years, with Iggy claiming in 2014 that her huge chart success was the only reason Banks was still getting any publicity.
In the early 2000s, rap feuds were all the rage. Nas and Jay Z. DMX and Ja Rule. 50 Cent and All Other Rappers.
But one of the most high-profile involved the biggest name in music at the time and the slightly less famous Benzino - a rapper and co-owner of hip-hop magazine The Source.
Benzino put out a track dissing Eminem called Pull Your Skirt Up, which described Eminem as a "2003 Vanilla Ice" and compared him to a woman.
Which is the ultimate insult, considering women are lesser human beings, right? (Apparently Benzino was born in 1859).
Eminem retaliated with his own diss track, Nail in the Coffin, in which he said: "Nobody wants to hear their grandfather rap."
Probably best to finish on one that has a happy ending.
RiRi and CiCi, as they should never have been nicknamed, both launched their careers in 2005 - with Ciara topping the charts with Goodies and Rihanna hitting number two with Pon De Replay.
They were the young bright hopes of R&B and even went on tour together. But a few years later things turned sour.
"I ran into her recently at a party and she wasn't the nicest," explained Ciara in 2011. "It wasn't the most pleasant run-in."
There then followed the most glorious of Twitter exchanges.
Rihanna: "My bad Ci, did I forget to tip you?"
Ciara: "Trust me Rihanna you don't want to see me on or off the stage."
Rihanna: "You gangsta huh?... Good luck with booking that stage you speak of."
Ouch. But it didn't take long for Rihanna to backtrack, writing: "Ciara baby, I love you girl! You hurt my feelings real bad on TV, that's why I retaliated this way. So sorry!"
Ciara graciously responded: 'Rhi, u know its always been love since day one! You threw me off in that party! Apology accepted. Let's chat in person."
Now isn't that lovely. Let's all be friends.
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Katy Perry has said she's ready to put an end to the bad blood which has existed between her and Taylor Swift for the past few years.