The name ``free probability'' does not refer to an attitude in the practice of probability, but to a theory initiated by D. Voiculescu about 15 years ago, with motivation from problems on free products of von Neumann algebras. The subject has evolved since then into a kind of parallel to basic probability theory, where tensor products and independent random variables are replaced by free products and free (non-commuting) random variables. The talk will avoid the von Neumann algebra side of the theory, and will focus on the latter parallelism (free-ness for random variables, as a non-commutative analogue for the concept of independence). For instance -- there exist free analogues of the Gaussian and of the Poisson distributions; the free analogue of the Gaussian is the semicircle law of Wigner, and appears as limit in the free central limit theorem of Voiculescu. A nice way to prove the free CLT goes by using the free counterpart of the notion of characteristic function, which is called ``the R-transform'' (of a probability measure on the real line). The list of free analogues could certainly be continued (e.g. free analogues of infinitely divisible distributions and of the Levy-Khinchin theorem, free stable laws, processes with free increments) and is a subject of current research. A notable feature of free probability is its connection to the theory of random matrices, which is already signaled by the occurrence of the Wigner's law in the free CLT, but also shows up in various other places.