Places to visit in London

I'm going to be in London for a few weeks and wanted to get some ideas of what to do when I'm there.

Other than seeing the usual tourist spots-i.e. the Palace, Big Ben, the Eye-I was hoping that you guys could recommend neat places that are not typical for tourists to check out.

Also if you have in mind of any events that are going on in the next few weeks, please mention.

Thanks for all you help!