Private Information Retrieval asks an interesting (to me) question:
Can you make queries from a database without the database knowing what you want ?
The answer is, of couse, that you can, but you have to be clever about it. This has a strong bearing on how Google and the like store our personal information (which an IP address is pretty well equivalent to). Related question: do most citizens notice the loss of privacy due to CCTVs? Question: do they notice the benefit when they need them?