Consider a morphism of schemes f: X--->Y essentially of finite type over a field. When Y regular and one dimensional, it is a classical result (which can be found already in Hartshorne's Algebraic Geometry, for example) that f is flat if, and only if, every associated point of X maps to a generic point of Y. I will present an extension of this result that holds over any smooth base Y, but imposes no restrictions on its dimension.