This article gives a concise survey on important problems of the theory of permanents. The definition of the permanent of a matrix and a summary of some of its basic properties are followed by a treatment of the problem of converting permanents into determinants, and a discussion of the van der Waerden conjecture that has been proved quite recently. Very important are lower and upper bounds for permanents; these are presented for (0,1)-matrices, nonnegative matrices, and (1,-1)-matrices. Finally, we give a brief review of a selection of other attractive topics in the theory of permanents, particularly very recent developments.