There are 3684030417 non-isomorphic semigroups of order 8 (according to entry A027851 of OEIS), among which only 5 are groups. The idea of finding a general classification of finite semigroups -- as was done with finite simple groups -- seems hopeless. But there are other ways to organize this mess, and a very nice one consists in sorting them into classes with nice closure properties, called pseudovarieties. Free profinite semigroups have turned out to be a very useful device to study these pseudovarieties. In my presentation, I want to talk about the free profinite semigroups, how they are defined, and what kind of properties they have.