The just-finished conference 4ICC, of which I was the chief organizer, produced some memorable talks, and even more memorable statements. Some of my favourites (paraphrased and from memory):

• Every romantic movie must have a kiss; every math talk must have a proof. But just as showing too much in a romantic movie leads to pornography, showing too many proof details is also in bad taste. (J. de Loera, also quoting H. Lenstra).
• There are two schools of statisticians with respect to this question. The first maintains that the question is well-posed and worthy of study, the second maintains that the question does not make sense. To which the first replies, “Yes, it does!”. That is as far as I have been able to penetrate into the subject. (A. Barvinok)
• If Santa Claus gives you the dimension, then you can solve it using only linear algebra. [description of how to do this] … The catch is, Santa Claus does not exist. I’m sorry to have to break it to you. (J. de Loera)
• “Cossing a toin”, a great Spoonerism from Doron Zeilberger