We introduce the distinguishing numbers of partially ordered sets. This study has given us the opportunity to apply classic results in poset theory to obtain results about this parameter. We present both an elementary proof of the distinguishing number of divisibility lattices and a more general result that employs Birkoff's Fundamental Theorem of Distributive Lattices. In other distinguishing bounds, we use a fundamental result about straight line embeddings of ranked planar posets with a minimum element and a maximum element to find bounds on planar posets.