Muzychuk drew widespread media attention at the end of 2017 for boycotting the World Rapid and Blitz Championships over FIDE's decision to hold the events in Saudi Arabia because of the country's discriminatory policies against women.

Their father originally taught them how the pieces move at a park where there was a large human-sized chess board on the ground.

Around the same time, Boris Kutin, a Slovenian who was the president of the European Chess Union, offered Muzychuk the opportunity to represent Slovenia, which she accepted.

[22][23][24] Her performance at the 2003 edition as well as a joint first-place finish at the 17-round Lviv's Hopes tournament later in the year helped Muzychuk reach a rating of 2300 by October 2003.

[20][32] One of her other best results in classical chess during these years was at the 2006 European Individual Women's Championship, where she scored 7½/11 and finished in equal third place.

[38][39] That summer, Muzychuk had her biggest triumph of the year by winning the World Junior Chess Championship for under-20 girls in Chotowa, Poland, finishing in clear first with a score of 11/13.

She produced her career-best result at the European Individual Women's Championship, scoring 8½/11 and earning a bronze medal.

[46][47] Muzychuk continued to perform well in the Grand Prix, finishing in joint first with Koneru Humpy at the Kazan event.

During the event, she became the fourth woman to reach a rating of 2600 after Judit Polgár, Koneru, and Hou Yifan.

[58] She had a better result at the Tata Steel Challengers tournament in Wijk aan Zee in early 2014, where she came in fourth place with a score of 8/13, corresponding to a performance rating of 2666.

[63] She stated her sister continuing to play for Ukraine as well as the new federation president Viktor Kapustin as reasons for her return.

[64] At the end of the year, she entered the Ukrainian women's national championship for the first time since she was the 2003 champion and again won the tournament.

[67] Early in 2016, Muzychuk won the women's first prize at the Gibraltar Chess Festival with a score of 7/10, corresponding to a performance rating of 2677.

[72][73] Muzychuk came closest to winning the classical Women's World Chess Championship in 2017 in Tehran, where she finished runner-up to Tan Zhongyi.

Exactly one year ago I won these two titles and was about the happiest person in the chess world but this time I feel really bad.

[79][80]Her announcement of the preceding comments on Facebook was shared over 74,000 times and received over 165,000 reactions predominantly in support of her statement.

[81] The following year in 2018, Saudi Arabia was scheduled to host the tournament again; however, it was moved to Russia less than a month in advance largely due to a separate issue of Israeli players being denied visas needed to compete.

[83][84] For the second consecutive Women's World Championship knockout tournament, Muzychuk faced Stefanova and Kosteniuk in back-to-back rounds, this time as the 4th seed in 2018.

[86] Although she had a strong finish, she could not overcome a slow start where she lost two of the first three games to Dzagnidze and Tan, and her final score of 8/14 only sufficed for second place.

Slovenia finished in the top ten three times in those four tournaments, including in 2007 in Heraklion when they came in 6th place out of 30 teams and outperformed their 11th seed.

[37] Since switching federations back to Ukraine, Muzychuk has played at three Women's Chess Olympiads and her team has medalled at all of them.

Individually, Muzychuk won the gold medal on the top board with an unbeaten 7½/10 and a performance rating of 2629, ahead of Hou Yifan.

[110][111][112][113][114] At the 2018 Batumi Olympiad, Ukraine were again the 2nd seed, using the same team except for swapping in Iulija Osmak on the reserve board.

They tied with China for first place with an unbeaten 18 points (+7–0=4), but ended up with the silver medal because of the tiebreak criteria.

Muzychuk narrowly did not earn an individual medal, finishing in fourth place on the top board.

Muzychuk won an individual silver medal with an unbeaten score of 6½/9 and a performance rating of 2621, behind only Alexandra Kosteniuk.

She switched to ŠK T-Com Podgorica in 2008 and then Ekonomist SGSEU Saratov, winning a team bronze and individual silver for the former club.

[122][123] Her teammates on CE de Monte Carlo regularly included Hou Yifan, Koneru Humpy, and Pia Cramling.

[128] Muzychuk has described her playing style as similar to that of Fabiano Caruana in that they both focus on preparing openings and are good at positional chess.

Anna thinks of herself as calmer and prefers spending time with family and solitary activities such as reading or watching movies, in contrast with Mariya who is more expressive and likes shopping and doing things outside of home.