Let f=∑anxn∈ℚ[x] be a power series which is also a meromorphic function in some neighborhood of the origin. The subject of the talk will be how certain conditions on f(x) as a meromorphic function actually guarantee that f(x) is an algebraic function of x. We shall begin by describing a common theme which runs through many such arguments. We then shall discuss a number of examples of such theorems (including a new such result).