The term `shadow' refers to the (weakly) holomorphic modular form which arises when one applies the ξ-operator to a harmonic weak Maass form. In many interesting cases, such a Maass form arises when a holomorphic but not modular generating series for some quantities of arithmetic interest is completed to a modular but not holomorphic form. For example, the shadow of Zagier's weight 3/2 completion of the generating series for Gauss class numbers is the simplest theta series of weight 1/2, and the shadow of the non-holomorphic Eisenstein series E*_2 of weight 2 is a constant (weight 0). In this talk, I will describe a systematic method for constructing such examples which accounts for many classical cases. This method can be generalized to Siegel modular forms. I will mention an interesting example for genus 2.