This outing marks the last leg of a journey that began three years ago: the Bridge Project, an artistic collaboration between the Old Vic, the Brooklyn Academy of Music and Sam Mendes’s Neal Street Productions. The object was to pair leading British and American actors in major plays, the Project being jointly run by Mendes and the Old Vic’s artistic director — one Kevin Spacey.

Now they come together, with Mendes directing Spacey in arguably the supreme Shakespearean role. Richard III is a part not lightly undertaken by even the finest of actors, but Spacey easily joins the illustrious company of the two other great Richards I have seen on stage — Antony Sher and Ian McKellen. He flits across the stage like a cockroach, twisted left leg in a calliper, left hand gripping a stick as he malevolently makes his bloody ascent to the crown.

In a radio interview, Spacey made an interesting historical point, that the creation of the role denoted “a young playwright who has yet to learn that you have to give your leading actor breaks”. Richard has three times as many lines as any other actor in the play and is indeed on stage almost continuously. There is an intimacy in his planning and conniving — as he often addresses the audience directly — that made me wish I had been closer to the stage than the upper (Lillian Bayliss) circle.

Mendes’s production brings out some of the elements of vicious humour in the play, with Spacey turning on faux campness that is cleverly done. His Machiavellian ambition is mirrored in a fine performance by Chuk Iwuji as Buckingham. But during a lengthy first half, as the lords come and go (often subtly executed with a brushing-close of their eyes and a light bulb switched off above them), their performances do not endure in the memory — despite a decision to highlight their and others’ names flashed up as chapter headings.

Haydn Gwynne as Elizabeth and Annabel Scholey as Lady Anne impressed in their scenes, but I was baffled by the way Gemma Jones’s Queen Margaret was portrayed.

There is clever business on video as Richard preens when asked if he wishes to be king; a less clever idea is to make ‘citizens’ in an early short scene handle-holding tube passengers (why even think of this?) Once Richard is crowned, Spacey throws away any light touch and becomes thoroughly nasty — and effectively so.

Expectations for this production have been understandably high. They are not quite delivered: this is 3½ star stuff overall (with Kevin Spacey at 4½).

Most performances are sold out. Some under-25s tickets are still available: please call box office on 0844 871 7628 for more information. Standing positions are available on the day of the performance.