After the release of her album I am..., Hamasaki became more relaxed with giving performances outside the country. After performing at the 2002 MTV Asia Music Awards, Hamasaki felt that by writing only Japanese lyrics, she was not able to bring her "message" to other countries. Realizing that English was a "common global language", she decided to include English lyrics in her songs. "Rainbow" contained three songs with English lyrics.
Hamasaki also worked with Dai Nagao and composed some songs under her pen name "Crea". The album was stylistically diverse from Hamasaki's previous works as it included rock- and trip-hop-influenced tracks as well as "summery", "up-tempo" and "grand gothic" songs and experimented with new techniques such as gospel choruses and the yells of an audience. The lyrics were also varied: themes in the album included freedom, the struggles of women, and "a summer that ends in sadness".
Three singles were released to promote the album—all of which became number one singles on the Oricon weekly charts. "Free & Easy" was the album's first single, and also Hamasaki's fourteenth number one single on the Oricon Weekly Charts. The track is said to be the result of a sudden inspiration Hamasaki and HAL had at an actual magazine photo shoot with Free & Easy magazine. The lyrics speak of "self-empowerment and freedom". Hamasaki said, "In Free & Easy, I wrote what I am feeling at the moment. In other words, I want to be a person who lives by taking responsibility for my own freedom, and I want to be a person for whom these words are perfectly applicable." The single also marked a dramatic change in the format of Hamasaki's singles. Singles after "Free and Easy" (with the exception of "No Way to Say") cut the number of remixes on the CD down to two or three, unlike previous singles with up to six or seven remixes.
The second single from the album was a multiple-A-side EP titled "H". It contained three tracks from the album - "Independent", "July 1st" and "Hanabi". The EP also reached number one position on Oricon charts and also became Hamasaki's sixth Million selling single and also made her the only artist to accomplish this feat. The single remains Hamasaki's last Million selling single to date. The lyrics of songs on "H" deal with different corners of life. Hanabi,a "reminiscent of the UK trip-hop/dub sound yet has a somewhat nostalgic feeling" deals with "a summer that ends in deep sadness". Independent is an anthem depicting "Independency" and July 1 is a "summer anthem", which featured guitarist Susumu Nishikawa.
The final single from the album was the Hamasaki and DAI penned "Voyage". The single had major classical influences with strings and violins. The song, like its predecessors, also topped the chart and also became one of Hamasaki's critically praised works. The single also garned the title of Hamasaki's longest charting single (with a total run of twenty-eight weeks). The single's music video became one of Hamasaki's two singles in which a short film was used in place of a regular music video (the other one being 2007's "Glitter / Fated"). The single was nominated for a Gold Disk Award.
The title track "Rainbow" that gave the album its title, was not included in the final track listing of the album and it was later included as the lead song in the ballad compilation A Ballads. However, initial pressings of the album included a password to a website to listen to an instrumental preview of "Rainbow" enlisted as the "00" track of the album. Also first versions of the song were used in the TV commercials for promoting the album.
Released in December 2002, Rainbow entered the Japan Oricon Albums Chart at number one, with about 445,000 copies sold in its first day. In the weekly chart, the album reached the top position, selling over one million copies. The album also topped the monthly chart and reached number two on yearly chart, behind Utada Hikaru's Deep River.
The album has sold about 1,857,870 copies in Japan and has been certified 2× Million by the Recording Industry Association of Japan (RIAJ). Since Loveppears, "Rainbow" became Hamasaki's first studio album that did not break the two million mark. Worldwide, the album has sold about three million copies. A digipak edition was also released featuring all four covers of the single to commemorate the sale of the millionth copy. Rainbow is the 89th best selling album in Japan of all times