The first floor is given over to the University of Liverpool's art collection - a teaching collection built by donation, bequest, and the accumulated taste of a university across a century and a half. Joseph Wright of Derby's portrait of Liverpool merchant Charles Goore stares out from one wall, all candlelit gravity. J. M. W. Turner's 1815 oil of the eruption of the Soufrière Mountains in St Vincent burns in another room - smoke and sulphur rendered in the painter's late, dissolving style. There are Lucian Freud nudes, an Elisabeth Frink sculpture, prints from the long British nineteenth century. The treasure for many visitors, though, is the cabinet displaying plates from John James Audubon's Birds of America - the double-elephant folio whose page-and-a-half life-size birds are among the most expensive printed books ever made. A copy was sold in 2010 for 7.3 million pounds. The University's pages turn periodically, so the wild turkey or the Carolina parakeet may be on view depending on when you visit.

The top floor is the Tate Hall Museum, named for the Liverpool sugar merchant Henry Tate (yes, the same Tate that funded the London gallery) who endowed teaching space here in the original building. The current exhibition is called Lightbulb Moments and is an unusual experiment: instead of organising objects by department, the curators have laid them out around the question of where ideas come from. A dental teaching skull sits near an archaeological cache. An oceanographic instrument shares space with a Victorian zoological specimen. Engineering models lean toward early medical apparatus. The premise is that good ideas come from cross-pollination - from noticing patterns one discipline cannot see alone - and that university collections, ramshackle and broad, are uniquely suited to making that argument. Visitors are invited to consider how their own ideas might come from spotting connections between seemingly unrelated things.

Waterhouse's building was originally University College Liverpool - one of three institutions (with Manchester and Leeds) that made up the federated Victoria University of Manchester. The name still resonates here. "THE ORIGINAL REDBRICK," the museum's homepage announces, claiming the bragging rights for the entire category. After Liverpool gained its own university charter in 1903, the building remained the symbolic heart of the campus even as the institution sprawled around it. By the late twentieth century it had become underused. The 2008 transformation kept Waterhouse's sequence of vaulted rooms - the tiled floors, the stone staircase, the carved capitals - and threaded the museum into them. The Waterhouse Café occupies the ground floor under the brick groins. The Leggate Lecture Theatre still holds talks, as it has since the 1890s. It is the rare museum where the building is as much the subject as the collection.