A power series F(T) is called hypergeometric if the ratio between the coefficient of T^n and the coefficient of T^{n+1} is a rational function of n. I will discuss the question of when such a series is also algebraic, i.e. p(T,F(T)) = 0 for some nonzero polynomial p. After considering some elementary examples, I will briefly describe the classification of all algebraic hypergeometric functions by Beukers and Heckman. I'll finish with a beautiful observation of Rodriguez-Villegas based on this classification.