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In this paper we present theory and experimental results on Algebraic Decision Diagrams. These diagrams extend BDDs by allowing values from an arbitraryfinite domain to be associated with the terminal nodes of the diagram.We present a treatment founded in Boolean algebras and discussalgorithms and results in several areas of application: Matrixmultiplication, shortest path algorithms, and direct methods for numerical linear algebra. Although we report anessentially negative result for Gaussian elimination per se, wepropose a modified form of ADDs which appears tocircumvent the difficulties in some cases. We discuss the relevanceof our findings and point to directions for future work.