Armstrong studied musical composition, violin and piano at the Royal Academy of Music where he was awarded the Charles Lucas prize and the Harvey Lohr scholarship for composition. He was also awarded the FTCL Fellowship in composition, and won the GLAA Young Jazz Musician of the Year in 1982. Upon graduation he became resident student composer for the London Contemporary Dance School. Upon completing his studies, Armstrong served as Music and Dance specialist at the Strathclyde Regional Council in 1984.

During the 1980s, Armstrong's composition work included commissions from the Arts Council for various classical ensembles in Scotland, and he also served as resident composer at the Tron Theatre in Glasgow. During this time he also received a Composer's Award to study electronic music, and maintained a parallel career in the Scottish Pop music scene, as a member of bands Hipsway, Texas and The Big Dish.

In 1990, his music theatre group Performance performed at the Glasgow Mayfest, and he composed a number of works for film, television, and stage productions by the Traverse Theatre, the BBC, and the Royal Shakespeare Company, among others. By the late 1990s he had gained critical acclaim for his work on the Baz Luhrmann films Romeo + Juliet and Moulin Rouge! (which won him a Golden Globe for Best Original Score in 2001). He has also done orchestrations with artists such as Hole, Massive Attack, Madonna, Spice Girls, U2, Suede, Shola Ama, Pet Shop Boys and The Future Sound of London. During this time he also released his first album of his own orchestral work, The Space Between Us, in 1998.

During the early 2000s Armstrong continued to produce film and orchestral work, including commissions for the Barbican Centre and the Scottish Ensemble. His most recent film scores are for Richard Curtis's 2003 film Love Actually, the Academy Award winning Taylor Hackford film, Ray (for which Armstrong won a Grammy Award) and Oliver Stone's 2006 film World Trade Center.

His track called "Escape" from the film Plunkett & Macleane is frequently used, since it is a rising crescendo type of music, on the British Television program "Top Gear" as the music which usually plays at the end of the "epic challenges" (for example, when Jeremy Clarkson completed a lap of the Nürburgring in 9 minutes, 59 seconds), and this track is commonly known, around the internet, as the "Top Gear Epic Music".

Starting in 2006 and continuing to this day, Armstrong's "Ball" track is used as the music in the filler/preview for test channel BBC HD in two edited forms and normally receives up to 10 airings a day.

In 2008, he wrote a fifteen minute opera for Scottish Opera, collaberating with popular author, Ian Rankin. The opera, titled 'Gesualdo', was one of five operas commissioned and clips from all five were shown as part of a programme about the future of opera on BBC1 Scotland. One of the other five operas was written by popular new composer, Gareth Williams.

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