The opening of the Burrell Collection in Pollok Country Park in October 1983 was a big deal for the people of Glasgow and beyond.

I recall visiting several times in that first year to take in its many splendours. For a young mind, it was dizzying to see so many disparate objects under one glazed roof. This idiosyncratic collection included everything from Chinese porcelain to medieval furniture, tapestries and stained glass to Degas pastels and a stunning bronze version of Rodin's The Thinker.

The Burrell Collection had been a long time coming. Sir William Burrell made his fortune in the family shipping firm, and when he and his elder brother sold almost their entire fleet in 1918, he switched his focus to expanding the art collection he had begun in his teens. In 1944, he pledged that, on his death, that collection would be placed in the hands of the City of Glasgow. He had a couple of provisos: namely that the collection should be housed in a rural setting away from the pollution of Glasgow and that it should not travel overseas.

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After he died (at the age of 96 in 1958), it would take 22 years for this legacy to be set in stone in the shape of the huge L-shaped, sandstone and glass modernist building designed by Sir Barry Gasson with Brit Andresen. Just 30 years on, this bold new gallery is already showing its age, as the swathes of tarpaulin and buckets dotted around the building testify.

An estimated £45 million revamp is required to set this A-listed building to rights. If funding is secured, it will close for an extended period, possibly between 2016 and 2020. Work will include a new roof, with the Heritage Lottery Fund expected to provide most of the funding. After the deluge of bad news about the Burrell, this is a welcome development, as is the recent decision by the Scottish Parliament to allow Burrell's bequest terms regarding the collection travelling outside Scotland to be overturned.

Ahead of these changes, the Bellini To Boudin exhibition has set up its stall, tarpaulin and all. Spanning five centuries, the well-known works on display include Bellini's panel painting Virgin And Child, a selection of works by Degas, a Rembrandt self-portrait from 1632 and Boudin's The Beach At Trouville, The Empress Eugenie. Other less familiar works include Whistler's Nocturne: Grey And Gold Westminster Bridge, and Portrait Of A Gentleman from the studio of Frans Hals.

This exhibition reveals as clearly as the crimson in Lucas Cranach the Elder's medieval masterpiece, Judith With The Head Of Holofernes, that Burrell's collection is a national treasure. This painting from 1530, the first one visitors see, depicts a triumphant Judith clutching a bloody sword after she has sliced the head off the despot Holofernes. The intensity of this painting sets the scene for the rest of Bellini To Boudin.

As you travel through the centuries via Burrell's painting acquisitions, it's fascinating to spot connections in his collecting. Burrell loved still lifes as well as fine figurative drawing and painting. The domesticity of some of the work links back to his compulsive acquisition of household objects, like beds, chairs and other furniture.

The Judgement Of Paris, painted about 1450-55 in oil and tempera by an unknown artist in Florence, once formed the side of a cassone - a wooden trunk given to a bride and groom at their wedding. Moving on, you discover that The Dog, painted by Jean Baptiste Oudry in 1751, was originally intended to hide a fireplace in summer months.

The information panels in this exhibition are a joy, and set out to tell the story of Burrell and his collecting process as much as the paintings themselves. Unlike other collectors who tried to create a chronological collection, he bought things he liked.

One which illustrates this point is Manet's The Ham, which is just what it says on the tin ... a ham. Manet described still life as the "touchstone of painting", and this beautifully restrained work was once owned by Degas, who is richly represented in this exhibition. The Burrell Collection has the finest collection of Degas art in Britain, and it's worth visiting just to see The Rehearsal, which is as fresh as the day it was painted.

Another cracker is Degas's unfinished Woman At Her Toilette, a portal into the way he worked, sketching quickly with charcoal and pastel on canvas. Other pastels to drool over are by Renoir, Manet, Gauguin and Pissarro. Hard to choose a favourite, but Gauguin's Breton Girl and Renoir's Lady With Auburn Hair stand out.

The show-stopper - at the tail-end of the exhibition - is a full-length portrait of Burrell's favourite sister Mary by Sir John Lavery, commissioned for her 21st birthday in 1895. With Burrell, as Bellini To Boudin reveals in spades, it was always personal.

Bellini to Boudin: Five Centuries Of Painting In The Burrell Collection is at the Burrell Collection, Pollok Country Park (0141 287 2550, until March 21, 2015