Fermat's problem, also called Fermat's last theorem, has attracted the attention of mathematicians far more than three centuries. Many clever methods have been devised to attack the problem, and many beautiful theories have been created with the aim of proving the theorem. Yet, despite all the attempts, the question remains unanswered. The topic is presented in the form of lectures, where I survey the main lines of work on the problem. In the first two lectures, there is a very brief description of the early history , as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in succession the main theories conneeted with the problem. The last two lectures are about analogues to Fermat's theorem. Some of these lectures were actually given, in a shorter version, at the Institut Henri Poincare, in Paris, as well as at Queen's University, in 1977. I endeavoured to produce a text, readable by mathematicians in general, and not only by specialists in number theory. However, due to a limitation in size, I am aware that certain points will appear sketchy. Another book on Fermat's theorem, now in preparation, will contain a considerable amount of the technical developments omitted here. It will serve those who wish to learn these matters in depth and, I hope, it will clarify and complement the present volume.

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In the first two lectures, there is a very brief description of the early history, as well as a selection of a few of the more representative recent results. In the lectures which follow, I examine in succession the main theories connected with the problem. The last two lectures are about analogues to Fermat's theorem.

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