Rational points on curves

I will discuss what is known and conjectured regarding the problem of determining whether a curve defined over the rational numbers has any rational points. In particular I will present recent examples that lead us to question some of the assumptions underlying a heuristic of Poonen supporting a conjectural solution to this problem. If time permits I will also explain the relationship with the Brauer-Manin obstruction. This is joint work with Bianca Viray and Felipe Voloch.