Built between 2001 and 2004 and officially inaugurated in 2005, La Grande Bibliothèque is the main branch of the Québec National Library and Archives (BNQ). It is visited by thousands of people every day (estimates vary, but recent numbers indicate over 12,000 visitors daily).
The building is worth a visit for its architecture (it was a team effort by Patkau Architects) - especially interior architecture - and as a quick-and-easy way to restore your faith in the next generation (presumably, the hordes of students devouring books, magazines, newspapers and other publications are all up to something good).
Access is free and you can take a self-guided tour by simply walking around each level. La Grande Bibliothèque also offers guided tours in English and French.