One of the most prolific bands in the pop and rock world,[lastfm link_type="artist_info"] Coldplay[/lastfm]  not only transcend genres with their note-defying, swooning instrumentation, lovelorn lyrics, and skilled songwriting, they are also a band with undeniable international appeal.

Once described as “Rock Statesmen,” Coldplay fans from places all over the world like Russia, South Africa, and Brazil tuned into their Live on Letterman performance tonight.

Watch Coldplay’s Live on Letterman performance right now on demand!

[cbsvideo]http://www.cbs.com/late_night/liveonletterman/coldplay/video/?pid=zsZsQJZuMoeDSeEP0o5jaVQVj_vZSL2Q[/cbsvideo]

Walking quietly onto the stage, Chris Martin shyly asks the audience, “Everybody okay?” before going directly into the jangly chime of the beginning of “Hurts Like Heaven,” a new song from Coldplay’s upcoming album Mylo Xyloto.  Bedazzled by a rainbow of laser lights cutting through circular grids and a piano covered with black-light graffiti, “Hurts Live Heaven” is an upbeat indie-rock song with a spirited vibe–perfect with the kaleidoscope of lights and art in the background.

Photo: Jeffrey R. Staab/CBS

Without hesitation and a flash of canary yellow lights , Coldplay went into “Yellow.” The song that ostensibly made Coldplay into an international alternative rock sensation, “Yellow” might be a song about unrequited love, but the sumptuous melancholy of the song almost reads as a supplication to a love newly-gained, or in the case of Coldplay, a profession of love for their newly-devoted audience.

Yellow makes me realise how much the've grown! Coldplay Live on Letterman on CBS.com: soc.li/IYGzsuK—
Sele Curcio (@selecurcio) September 21, 2011

Asking again if “everybody” is “okay,” with a cheeky smile Coldplay goes into the percussive and poignant, “In My Place” which starts with a highlight on Will Champion’s drumming and a plucky guitar rhythm from Buckland. Apparently a collaborative song-writing effort, all the different instrumental influences are apparent in the well-developed structure of the song.

During the middle of the song, Martin announces Buckland, calling him his best friend, and ushering him to the front of the stage so the crowd could give him his due praise.

Covered in sweat, Martin utilizes his quiet time on stage to recognize and appreciate the  super close crowd:  “I hope we have, by the end of the evening, a connection between band and audience.”

With his usual self-effacing sense of humor, Martin assured the audience that they “always look skinnier on television” and “haircuts always look as planned, even this receding thing,” he said, pointing to his head.

Photo: Jeffrey R. Staab/CBS

The band then went into “Major Minus” and “Paradise.”  Although Chris Martin drops the title to one of Coldplay’s other songs, “Every Teardrop Is A Waterfall” in the song “Paradise,” “Paradise” is more leisurely, yet anthemic old-school Coldplay ballad with viscid strings, expansive guitars, and piano-driven lyrics.

Martin cited [lastfm link_type="artist_info"]Blur[/lastfm] as a big influence on “Paradise,” which can be heard in the sweeping, evocative chorus.

Who is From London??? Coldplay Live on Letterman on CBS.com: soc.li/IYGzsuK—
Dan Simeí Castro (@WalkOnTheLife) September 21, 2011

Almost catapulting into the crowd, Martin asks the audience not to “be shy.” Starting with the urgent pull of bows across strings, “Viva La Vida” is one of Coldplay’s most extensive storytelling songs; even the title, meaning “Long Live Life” in Spanish tells its own story since it’s based on a Frida Kahlo painting of the same name. Deftly, “Viva La Vida” jumps from grandiose instrumentation to gossamer vocals.

Photo: Jeffrey R. Staab/CBS

Exhausted by his animated dancing, at the end of “Viva La Vida,” Martin melted down onto the stage, spent from his spirited performance.

A new song from Mylo Xyloto, “Charlie Brown” may be named after the tentative Peanuts character, but the song doesn’t hesitate to become a stage-scorcher. “Charlie Brown” is a sexy, uptempo rock song with an almost ’80s dance-jam sensibility and sparkling lyrics like “all innocent and wild, we’ll be glowing in the dark.”

Y se fue In my place! Ahora Chris se desabriga y habla, Will toma agua y se escucha un silencio tremendo! … soc.li/IYGzsuK—
Caro Botta (@carola_613) September 21, 2011

Much to the excitement of the crowd (and probably a much-needed break for the sweat-covered Martin), Coldplay launched quietly into “The Scientist,” a beautiful piano-driven ballad that seems to clamor sadly through the theatre like a wayward shooting star.

“The Scientist” acts as a perfect mid-set reminder of the painful pursuit of love and showcases Martin’s skills as a songwriter perfectly.

Photo: Jeffrey R. Staab/CBS

Centered around Martin’s dynamic piano-riff, “Clocks” is the quintessential “festival” song, but with soft, tender moments in the sound that read as pure intimacy. Buckland’s gorgeously-structured guitar chords are subtle enough to be complimentary to Martin’s compelling choral entreaty of “You are.”

Escuchar Coldplay me lleva a mi infancia. Aquellos tiempos :')—
Veronica Montoya (@_vamc) September 21, 2011

Awash with a greyish-blue haze, Coldplay looked comfortable singing the profound “Fix You,” even though Martin asked the audience, again, if they were “okay.” From Coldplay’s 2005 album X&Y, “Fix You” might have been around for six years, but the emotionally evocative songs is still one of Coldplay’s most highly praised.

Rumored to be written as a memorial for Gwyneth Paltrow‘s deceased father, Bruce Paltrow, “Fix You” features lush keyboard fingering, melancholic but hopeful lyrics, and Martin’s pained, yet beautiful falsetto. Almost to break up the painfulness of the song, Martin ended the song by going into a deep, humorous bass.

Photo: Jeffrey R. Staab/CBS

“Thank you so much for giving up your Tuesday. And thank you everyone on the web for giving up some of your Tuesday evening too,” says Martin before he croons “I feel my heart start beating to my favorite song,” in “Every Teardrop Is A Waterfall.”

@coldplay i thank each of you for taking time to get it 'right'. THANK YOU!!! cheers and rock on!!! love it all!!!—
jeff cofer (@jwpeppa) September 21, 2011

The song starts into its bombastic, beat-driven first verse. Martin gets so into the song that he falls over onto the stage, but immediately rolls back up onto his feet and keeps bouncing with just as much enthusiasm.

With a voluminous electro-pop keyboard melody, U2-esque drums, and a blithe upbeat bounce, “Every Teardrop Is A Waterfall” is both a perfect way to close out their set and a definitive tune for the future sound of Coldplay.

As per usual, Coldplay’s audience broke into a steady, heart-warming applause and wouldn’t let Coldplay go until they did an encore. Martin came out with an acoustic guitar and said, “Are we still on the web because we slightly flugged this bit up…we weren’t going to do an encore.”

“We thought you’d be a slightly colder audience. So, just give us thirty seconds to get our s**t in order…We have one more song. It’s new, but it isn’t that bad. This is how we advertise all our songs.”

Solo Chris la cantara con su guitarra y sin microfono!! #LiveOnLetterman—
Coldplay SLP (@ColdplayMX_SLP) September 21, 2011

Martin was having trouble with his mic, so he went off mic for a second before he went into a moving acoustic rendition of one “Us Against The World.”  Almost evoking a garage jam session, the rest of the band came out in the middle of the song to lead their instruments and voices.

In the end, it was just Martin singing a mantra softly to his audience, “through chaos as it swirls, it’s just us against the world.”

And in that moment, we know that Coldplay truly connects with their audience on more than just a musical level.

I LOVE YOU TOO CHRIS <3333333. Coldplay Live on Letterman on CBS.com: soc.li/IYGzsuK—
Celine A ▲ (@troubleMX) September 21, 2011

The “love” is mutual.