  This is an extended version of an invited lecture I gave at the Journees
Arithmetiques in St. Etienne in July 2009.
  We discuss the state of the art regarding the problem of finding the set of
rational points on a (smooth projective) geometrically integral curve C over Q.
The focus is on practical aspects of this problem in the case that the genus of
C is at least 2, and therefore the set of rational points is finite.
