A handful of times in operatic history, a composer has thrown away the rule book and produced a work which defies categorisation because it simply isn't like anything else in the repertoire. Kurt Weill's Die Dreigroschenoper is one such work: musically and dramatically, neither Weill himself or anyone else has ever written anything remotely similar.

It helps when your librettist is one of the great ground-breaking dramatists of history. The "Threepenny Opera" is Berthold Brecht's response to the financial and political catastrophe that enveloped Germany in the 1920s. Brecht was very young and very angry: the drama proceeds at an exuberant pace, the humour is blacker than black, the language is biting, sardonic and riotously funny. As in many of his works, Brecht took an established piece, John Gay's 18th century The Beggar's Opera (itself a fine satire on both the politics of its day and the conventions of Italian opera) and twisted and updated it to his own theatrical and political ends.

But Weill made Die Dreigroschenoper into something far more than a vehicle for Brecht's political agenda. Where most operas, even the great ones, have only a tiny number of really memorable numbers, Die Dreigroschenoper brims with highlights: I can count a dozen (to include the famous pair of The Ballad of Mack the Knife and Pirate Jenny). The musical style is made unique by the sheer level of melodic invention: several of the numbers are made of several melodies woven together (one in the voice, others in instrumental accompaniment), the harmonic progressions are rarely simple and the rhythmic styles are borrowed from everywhere from church chorales and Beethoven sturm und drang to romantic melodrama, jazz and tango.

First prize of the evening goes to the production team. Director and co-set designer John Ramster clearly took to heart Brecht's trademarked principle of Verfremdungseffekt (translated as "defamiliarization" or, misleadingly, "alienation") in which the dramatist uses a series of techniques to take the audience out of its comfort zone and render it more susceptible to surprise and wonderment. We were continually on the receiving end of small surprises: blue flashing lights on policemen's helmets, a large silver lamé pound sign on the long skirt of Jenny the whore, a pastiche Broadway showtune dance of Macheath and his captors (complete with ostrich feathers), an Egyptian headdress worn by Peachum as he warns the police chief Tiger Brown of the fate of his equivalent in ancient Egypt, or any of the brilliant pieces of choreography for the ensemble numbers. The cast kept up a blistering speed throughout the evening, and their acting quality shone: every cast member gave us crisp, telling renderings of their characters (in many cases, several each; eight performers played a total of a couple of dozen parts).

Surprisingly, in view of Brecht's reputation, the politics of much of the piece are rather muddled. The overall message is that "the human condition is harsh, we should be swift to help and slow to judge," but it's wrapped in a collection of highly unsavoury characters. The most powerful point was the finale of Act II, where the whole ensemble stood bolt upright to lecture the audience that when push comes to shove, "food comes first, morals later". It was done superbly.

Vocally, the undisputed star was Katie Bray as Polly (who was also given Pirate Jenny to sing, transplanted rather oddly into her wedding scene). Bray gave us wonderful feel for Weill's melodic lines and for every mood, whether lyrical in her Farewell Song, waywardly self-willed in the Song of Yes and No or rapid-fire in her Jealousy duet with Runette Botha's Lucy; Botha produced the other show-stopping performance of the evening in the Fight over the property. Adam Marsden's rendition of Tiger Brown was the other big vocal highlight: he has a powerful, rich bass-baritone voice which one could listen to for hours. Stephen Aviss has a lovely lyrical voice: he sang Macheath's Call from the grave wonderfully, but he didn't always have the sneering self-confidence to make the most of Macheath's music, and was badly overpowered by Marsden in the Cannon song duet.

Weill's music is more difficult to play than it appears: there aren't many instruments (in the original performance, the many instrumental parts were shared amongst just eight musicians), and there are a large number of different styles to get your head around. The band, conducted by Dominic Wheeler, got it fairly right most of the time, but I've heard it played with somewhat more edge and bite on other occasions.

In my view, Die Dreigroschenoper is one of the greatest works of musical drama ever written and deserves to be performed far more often than it is. Perhaps it's because opera diehards consider it "not really opera" and lovers of straight theatre or musicals consider it "too operatic". But when it's done to the excellent overall quality that the Royal Academy managed last night, it's hard to beat.