Abstract: Reading from a dictionary, we are told that a paradox is any statement contrary to common belief or any statement that seems contradictory, unbelievable, or absurd but that may in fact be true. We will look at some mathematical ideas from antiquity to the present day which have been called paradoxical and try to determine in each case whether or not there is really anything mysterious going on. While we may discuss a smattering of the some of the set theory which lies behind some of these ideas, this talk should be very accessible to all.