[18] It is an urban Europop song with "strong synth sound" and "African tribal beat",[19][better source needed] penned by Yoo Young-jin and composed by Will Simms, Chad Hugo (The Neptunes), Chris Holsten, and Anne Judith Wik (Dsign Music).

The original music video for "Happiness" gained more than two million views on YouTube in its first 24 hours of release before being removed due to controversy over problematic background images and replaced with an edited version.

[22] The song, which includes a rap verse from then-SR14B member Taeyong, was a remake of the 2000 single of the same name by S.E.S., SM Entertainment's first girl group.

[23] The music video for the song was directed by Kwon Soon-wook and Shim Jae-won, choreographed by Kyle Hanagami, and featured original choreography seen in a pre-debut SM Rookies teaser of Irene and Seulgi.

[32][33] On March 19, the group held an album showcase on Ice Cream TV, a program streamed through Naver Music and hosted by Shinee's Minho.

[35] In August 2015, Red Velvet held their first US performance during the annual KCON convention and music festival in Los Angeles, California.

[37] Billboard's Jeff Benjamin praised The Red "an impressive, solid debut album", stating that it "indicates big things for the act that needs to follow in the footsteps of their beloved female label mates Girls' Generation and f(x)".

[42] Red Velvet went on to take part in the SM Entertainment's special Christmas project, Winter Garden, releasing a digital single titled "Wish Tree" on December 18, 2015.

[59] From July 27 to September 10, Red Velvet starred in their first reality television program, Level Up Project, which showcased footage of their trip to Thailand.

[64] In August 2017, Red Velvet's rendition of Yoon Jong Shin's "Rebirth", made possible by the show Snowball Project, was officially released as part of SM Station.

[67] The Red Velvet Premium Showcase F'U'N Room Reveluv-Baby Party took place at the Yebisu Garden Hall in Tokyo on November 6.

This was further cemented when Perfect Velvet sold over 100,000 copies and "Peek-A-Boo" won first place on the music program Inkigayo on their last day of promotions for the song, proving its longevity on the charts.

[94] Its lead single "Power Up" earned the group a "perfect all-kill" for the first time in their career when it topped all of South Korea's real-time, daily and weekly music charts upon release.

They held shows in Los Angeles, Dallas, Miami, Chicago and Newark in the United States and Toronto and Vancouver in Canada, becoming the first K-pop girl group to hold a North American tour in three years.

[103][104] On April 5, Red Velvet were featured on a remixed version of "Close to Me" by Ellie Goulding and Diplo, with members Yeri and Wendy contributing Korean lyrics to the song.

Commercially, the CD version of the EP debuted atop the Gaon Weekly Album Chart, and sold 156,993 copies by the end of 2019 in South Korea, placing thirty-seventh for the year.

[117] In early 2020, La Rouge continued as an arena tour in Japan without Wendy, who was injured in a stage accident during rehearsals at the 2019 SBS Gayo Daejeon on December 25.

[132] By March 20, 2022, it was reported that The ReVe Festival 2022 – Feel My Rhythm accumulated 516,866 total pre-orders, a career high for Red Velvet, making them half-million sellers.

As of April 2022, The ReVe Festival 2022 – Feel My Rhythm sold 653,610 copies on the Gaon Chart, making it the group's best-selling physical release of their career.

[145] In March 2023, SM announced that the group's fourth concert tour, R to V, would start in early April at KSPO Dome in Seoul.

[149] On December 31, 2023, Red Velvet members Irene, Seulgi, and Wendy performed in the Philippines for the New Year's Countdown held in Bonifacio Global City (BGC), Taguig.

[154] On July 5, SM announced that there would be an additional performance on August 2 in Seoul for the fancon tour due to selling out of all tickets for the original dates.

[155] On July 15, it was announced that Red Velvet would be releasing a new version of Cosmic on August 1, the day of their tenth anniversary, with the addition of a new fan song "Sweet Dreams".

[175] Red Velvet have been lauded for breaking stereotypes among popular girl groups in South Korea, who tend to fall under either "cute and pure" or "sexy".

In a country where girl groups' fanbases are mostly male, Taylor Glasby of Dazed Digital noted that the majority of Red Velvet's fans are young women.

[176][177][178] IZE named the group as one of the successful female figures who helped transform the "passive image" of South Korean women in 2017.

[180] Janelle Okwodu from Vogue wrote that throughout their career, "the critically acclaimed South Korean girl group has dominated the charts with their innovative pop sound".

[181] Billboard reported that Red Velvet were the overall favorite K-pop group of 2018 among fans of every gender and sexual identity on the genre-specific community on popular internet forum Reddit.

[182] Red Velvet have topped the Girl Group Brand Power Ranking, published by the Korean Corporate Reputation Research Institute, several times.

[189] Discussing the Korean Wave in 2018, the director of the Korea Foundation for International Cultural Exchange cited Red Velvet as a major contributor and one of the country's most talented idol groups who have "largely promoted K-pop" around the world.