There's the Uffizi Gallery in Florence, which pairs a unique building (designed by Vasari) with an unmatched collection. The huge Botticellis are great, but I especially enjoyed Michelangelo's Holy Family, a much smaller work that wasn't surrounded by gawkers.
Likewise, even the second-rate stuff at the Galleria Borghese in Rome is fascinating. But the main draw is the Bernini collection, a must for any fan of Mannerist or Baroque sculpture.
In Brussels, there's the Centre Belge de la Bande Dessinée. The Victor Horta building alone is worth seeing, though the collection of comics is of course tough to pass up too.
It's harder to find excellent collections of European art in America, but what the US lacks in artistic tradition, it makes up for in art-hungry billionaires. Hearst Castle – the inspiration for Citizen Kane's Xanadu – has lots of loot in it, though the magnificent buildings and site developed under Hearst's direction are the real attraction.
The Getty Center, featuring the collection of oil magnate J. Paul Getty, also has an impressive amount of European art. But the stunning views of Los Angeles from the center's buildings are probably the most memorable part of any visit.
Obviously, this list could have other museums too. I'll add others if these articles prove popular.