Generalized down-up algebras, introduced by Cassidy and Shelton in 2004, is a certain deformation of the enveloping algebra of sl(2). It encompasses many previously studied algebras, including the down-up algebras of Benkart and Roby (of course), Witten´s seven-parameter deformation of U(sl(2)), and certain subalgebras of the quantum group U_q(sl(3)). Generalized down-up algebras have small enough dimensions so that direct calculations yield satisfactory information, but rich enough so that interesting phenomena appear. For example, it is possible to determine all primitive ideals of generalized down-up algebras without sophisticated machinery. I will provide a reasonably self-contained illustration of such a calculation.