The purpose of this expository monograph is three-fold. First, the solution of a problem posed by Wolfgang Krull in 1932 is presented. He asked whether what is now called the &39;Krull-Schmidt Theorem&39; holds for artinian modules. A negative answer was published only in 1995 by Facchini, Herbera, Levy and V&225;mos. Second, the answer to a question posed by Warfield in 1975, namely, whether the Krull-Schmidt-Theorem holds for serial modules, is described. Facchini published a negative answer in 1996. The solution to the Warfield problem shows an interesting behavior; in fact, it is a phenomenon so rare in the history of Krull-Schmidt type theorems that its presentation to a wider mathemati