It's been interesting to note, as the museum developments of the Great Lottery Years rolled inexorably on, the extent to which it has predominantly been architects who have been praised or blamed for a project's subsequent success or failure. In the case of the recently reopened Ashmolean, exhibition designer Metaphor, aided and abetted by museum director Christopher Brown, may well turn out to be the real hero of the piece. Yes, architect Rick Mather has opened out the 1840s CR Cockerell building into something unashamedly contemporary and respectfully timeless, but it's the apparent transformation in the museum's internal culture, at least as reflected in the approach to interpretative display, which is the most remarkable aspect of this project for me.

Despite this, cynics and observers of the inner workings of university museums may need stronger evidence of real change than that elegantly afforded to the public by the deliberately fluid new Crossing Cultures. Crossing Time theme. For much of its complicated history, the Ashmolean has been content, and perhaps even proud, to rest on the laurels of its superlative collections and their related academic fiefdoms. Despite some good work undertaken from time to time by its education staff, there was an uncomfortable sense that academic visitors weren't entirely welcome and that if you didn't already know about the objects on display, this wasn't really tube place for you. Such endemic attitudes can be hard to shift, but in terms of the public encounter with the new Ashmolean, that arrogance of approach has to all intents and purposes vanished. It begins with the reopening of the central main entrance under the podium. For years, public access to the museum was from the lower levels to the side of the grand steps, so there was already a sense of being allowed in by the servants' entrance even before you'd begun making sense of the museum's baffling layout and alarming departmental inconsistencies. Now, the visitor is drawn across Cockerell's skinny original building into Mather's airy atrium. Which itself leads into the 39 new galleries that have provided 100 per cent more display space. The new galleries are in the area formerly occupied by the jumble of infill, created in the 19th century and later. That crowded behind the Cockerell facade. They are a triumph.

Accessed from the full-height staircase that skims the atrium walls, the five floors of windowless galleries - arranged using broad categories such as West Meets East, Asian Crossroads and Ancient Worlds - are rich with wonders. The demolition of the old lines of demarcation on the basis of object type has allowed for an exploration of cross-cultural influence that, despite a vestigial tendency towards the Eurocentric, allows you to make really exciting connections. The text is clear, helpful and seldom condescending. The very largest cases can mean that you sometimes feel physically and intellectually distanced from the objects, but there's also a nice device of large central cases running the length of galleries with objects presented at varying heights, which means that a lot can be packed in without an effect of serious overcrowding. The Tradescants' collections of "curiosities", which were at the heart of Elias Ashmole's founding bequest, used to be tucked away at the side of an upper floor. Now they're gathered in a modern-day cabinet that is effectively at the heart of the museum. On each floor there's an introductory gallery which supports the Crossing Cultures, Crossing Time principle by suggesting themed approaches to thinking about the world and the collections. It's almost inevitable, given the need to present so many truly iconic works, that this system sometimes collapses back into familiar departmental territory, but on the whole it all works surprisingly well. The contrast between the old and new buildings, felt by some commentators to be cold and unfeeling, didn't bother me. There is no attempt to be anything other than open and there seems to be a genuine desire to exploit the opportunities afforded by taking the old Ashmolean apart. More unsettling is the contrast between the new galleries and those remaining unreconstructed in the original Cockerell building.

The Arundel marbles, statues and casts that file down the ground floor gallery are of enormous significance in the histories of collecting and the museum, but unless your Latin is up to scratch, the monumental inscriptions leave you none the wiser. There are plans to change this, but in the current economic climate when funding for university museums is under threat, such schemes are necessarily contingent on cash flow.

Reservations? Supporters of the old order will be relieved to know that, with the presiding genius of Apollo in the centre of the atrium, the gods still reign at the top of Olympus. Climb to the top of the building and you'll find that the highest and best is reserved for the director's office and the boardroom, with mere mortals granted temporary access to heaven in the rooftop restaurant. Then there's the question of the spaces allocated to education. Relatively small, with limited daylight and in a half-basement, their place in the order of things feels disappointingly clear, despite institutional protestations to the contrary, particularly in light of the Whitechapel Gallery's wonderful new provision.

Overall, this is a real success story. There will have been a few academic staff at the Ashmolean who needed to be dragged kicking and screaming through the 19th and 20th centuries into the 21st and it is to the enormous credit of the directorate and the designers that, as far as the visitor is concerned, such seismic change has been achieved with no blood evident on the carpet.

If everyone can hold their nerve through the impending fund crises, then Oxford will be at boast not only of the Ashmolean's great collections but, at last, of a great university museum that properly honours them.