I was searching for a museum in Central Belfast (there isn't one, you need to go south towards Queens University to find it) and found this place after asking at the Tourist Information Centre. It is a pleasant surprise. Not only is it a libary but it's also an exhibition space, and a mine of information about Ireland in general and Belfast in particular.

Good cafe and clean toilets too. :-)

This is a wonderful calm haven in the centre of Belfast. The Cafe is always peaceful, the food excellent and the staff are really pleasant and helpful.

Coffee aside, this little gem, right in the middle of Belfast, is a civilised haven of books (and not just Irish ones) and everyone who works there seems to have time for your inane questions

My daughter became a member of The Linenhall Library & I also joined. (It is a subscription only library, unlike the other public libraries which are free). It is the oldest library in Belfast and is a mine of information about Ireland in general and Belfast in particular. There are countless books and newspapers about what is known as that awful period in our history here in Irleand which has become known as 'The Troubles'. At the last count, I was told that they hold in excess of 250,000 different items relating to this period alone, in the Northern Ireland Politicial Collection.
They also have an excellent genealogy section, for those interested in doing some family history research and the amount of information here could prove invaluable.
There is an excellent free exhibition on until Christmas Eve in the Vertical Gallery which is showing posters documenting the years of 'The Troubles'. Worth seeing and you will find the Linenhall Library an oasis of calm in an otherwise bustling, busy Belfast city centre. (Library is opposite Belfast City Hall near corner of Fountain Street)
Enjoy!