The Farey sequence of order n is the ordered sequence of irreducible fractions whose denominator does not exceed n. Despite this simple definition, the Farey sequence takes part in very rich mathematics. Indeed, consecutive terms in the Farey sequence satisfy arithmetic and combinatorial relations that make this sequence the link between many a priori unrelated topics. This is especially true in geometry. In this presentation I will illustrate some of the frequent and elegant appearances of the Farey sequence in geometry, in particular in hyperbolic geometry and in flat geometry.
