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This monograph reviews a class of univariate piecewise polynomial functions known as discrete splines, which share properties analogous to the better-known class of spline functions, but where continuity in derivatives is replaced by continuity in divided differences. As it happens, discrete splines bear connections to a wide array of developments in applied mathematics and statistics, from divided differences and Newton interpolation, dating back 300 years, to the recent emergence of trend filtering. In this concise yet comprehensive monograph, the author uses his recognized expertise on the subject to guide the reader through these connections. In doing so, the author provides an insightful journey through the historical and most recent developments, contributing some new perspectives and results along the way. Written for researchers and advanced level students of applied mathematics and statistics, this monograph will be of particular interest to those using trend filtering in machine learning applications.