By definition, a smooth complex projective variety is rationally connected if any two points on it can be joined by a rational curve. I will discuss the class of rationally connected varieties and how its behavior compares favorably to the more familiar notions of rational or unirational varieties. In particular, I will discuss joint work with Joe Harris and Jason Starr which shows that this class of varieties is closed under formation of fibrations and how this fact is relevant to the problem of classifying higher dimensional algebraic varieties.