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If someone told you that mathematics is quite beautiful, you might be surprised. But you should know that some people do mathematics all their lives, and create mathematics, just as a composer creates music. Usually, every time a mathematician solves a problem, this gives rise to many others, new and just as beautiful as the one which was solved. Of course, often these problems are quite difficult, and as in other disciplines can be understood only by those who have studied the subject with some depth, and know the subject well. In 1981, Jean Brette, who is responsible for the Mathematics Section of the Palais de la Decouverte (Science Museum) in Paris, invited me to give a conference at the Palais. I had never given such a conference before, to a non-mathematical public. Here was a challenge: could I communicate to such a Saturday afternoon audience what it means to do mathematics, and why one does mathematics ? By "mathematics" I mean pure mathematics. This doesn't mean that pure math is better than other types of math, but I and a number of others do pure mathematics, and it's about them that I am now concerned. Math has a bad (!) reputation, stemming from the most elementary levels. The word is in fact used in many different contexts. First, I had to explain briefly these possible contexts, and the one with which I wanted to deal. Many people raised questions on a variety of topics: pure math, applied math, concrete versus abstract, the teaching of mathematics, and others which gave rise to a lively dialogue. But mostly, I wanted to show what pure mathematics is by examples: by doing mathematics with the people in the audience. And not artificial or superficial mathematics, but real mathematics, recognized as such by real mathematicians who do research in mathematics. So I had to find some topics which on the one hand were accessible to the Saturday afternoon public, who doesn't want to get bored or snowed, but who wants to learn without having any particular background in mathematics. That's what I did the first time, in 1981. It worked so well that I came back twice after that, in 1982 and 1983, each time choosing a different topic. The first two are rather algebraic: prime numbers and diophantine equations: while the third is geometric: great problems of geometry and space. The third time was a true marathon, during which a hundred persons stayed more than three and half hours ! I had never expected that one could achieve such a result. I was extremely touched by the audience reaction, all three times, but especially this last time! This book reproduces the three talks I gave in Paris. They were transcribed from tapes as faithfully as possible, so as to keep the lively style. Originally, they were published in the Revue du Palais de la Decouverte (the journal of the Science Museum). Who is Serge Lang ? What does a mathematician do and why ? Prime Numbers. A lively activity: To do mathematics Diophantine equations. Great problems of geometry and space