The latest official royal portrait, and the first painted portrait featuring the Princes William and Harry, hangs in a small room at the National Portrait Gallery among a selection of royal portraits of the Windsors. There’s the rather quirky one of the Queen Mother, painted in 1989 by Alison Watt, an artist who sought to capture her sitter “as ordinary as possible”. What our attention seems most drawn to is the china cup turned upside down on the arm of the Queen Mother’s armchair. Eh?

Walk into the gaping mouth of the metal container featured in Miroslaw Balka’s installation at Tate Modern’s Turbine Hall and you are plunged into a disorientating darkness. Unnerved, you shuffle forward, passing and perhaps finding comfort in the ghostly presence of other limbs, other bodies which are also shuffling uncertainly, all awareness of spatial relationships denied in the enveloping blackness.

Perhaps we think we’ve got the whole thing more or less sewn up in the nurture versus nature debate. DNA profiling, gene studies, twin studies, inherited traits - this is the stuff we read about almost daily and it is all meant to tell us who we are. At any rate we seem to live in a culture obsessed with genealogy, which is perhaps as much to do with living as atomised units as it to do with the latest research about genes, or what used to be called heredity.

If Andrew Graham-Dixon's arts career ever goes belly-up, there is surely a microphone with his name on it at Radio 4, so warm and confident and trustworthy is his voice. Judging, however, by his new three-part programme on BBC Four, The Art of Russia, there is no chance of this happening soon.

We watch and listen simultaneously so much today that it hardly seems blasphemous for a superlative pianist to decide to conceive an evening of piano music plus video installation. Leif Ove Andsnes has doubts about the transmittability of classical music to a general audience today - he calls the status quo into question, and he may be right. So he turned a concert programme into a video show, focusing on Musorgky’s Pictures at an Exhibition and Schumann’s Kinderszenen, to which would be set a visual installation around him and his piano.

Though he has yet to make a perfect film, the director Tim Burton’s choice of Gothic and fantasy subjects and his deadpan, post-expressionist approach to them rightfully designate him an auteur of considerable genius. His 14 movies to date have earned him a cohesive retrospective at New York’s Museum of Modern Art.

A 15ft aardvark constructed from raw timber with a light-up robotic face and gigantic hands is climbing up one of the support pillars of the Westway, next to the body of a full-sized helicopter the front of which has been shaped into a grinning skull. Life-size rearing horse torsos made of white marble-like resin, with real horse skulls instead of heads, are mounted on the wheels of Victorian perambulators, while a man rides a clanking, hissing, fire-spitting motorised beast with stamping front legs and huge rear wheels around through the crowd as children caper about and their parents drink rum punch to the sound of militant 1970s funk. It's a bit livelier than White Cube, that's for sure.