In his celebrated paper in 1981, Lempert introduced extremal discs or complex geodesics for the Kobayashi metric on a convex domain D in complex space. An analytic disc f: Delta to D (where Delta is the unit disc in the complex plane) is called extremal if |f'(0)| attains its maximum among all analytic discs in D with the same values and directions at 0 in Delta. Extremal discs are important biholomorphic invariants of D. Their boundaries lie in the boundary bD of the domain and form a family of curves invariant for the CR structure of bD. Lempert applies extremal discs to the study of biholomorphic mappings (a local version of Fefferman's theorem), Monge-Ampere equations, normal forms of convex domains, and other questions. Bland and Duchamp apply extremal discs to deformations and embeddibility of CR structures. There are other applications. We introduce extremal discs for CR manifolds of higher codimension.