Stable moments associated with pinned walls in a multidomain (MD) grain can be screened by regions in which walls are free to move. This screening problem is investigated by using a new approximation method that linearizes the demagnetizing field acting on individual domain walls in a grain. The approximation is shown to be usually excellent by comparing results obtained from it to more precise numerical solutions. In addition, the approximation enables one to carry out analytical calculations. Screening in a MD grain is generally very complex, and it was argued by MOON and MERRILL (1986a) that it might even lead to a self-reversal of a grain in some cases. However, it is shown in this paper that the particular domain state required for such a self-reversal will not likely ever be realized in practice. Moreover, it is found that the screening effect cannot explain the decrease of partial thermoremanent magnetization (pTRM) on cooling that is sometimes observed in MD magnetites; instead some other process such as a transdomain process is likely involved.