The continued fraction expansion of a real number is a very efficient tool for finding its best rational approximation. Moreover, continued fractions occur both in many theoretical questions in number theory, complex analysis, dynamical systems, as well as in more practical questions related for example to cryptography. After giving an introduction to the classical theory of continued fractions in the field of real numbers and to famous results and open problems in this setting, I will explain how this theory can be generalized to other worlds, for example to the field of p-adic numbers, where many differences with the classical case arise.