This book describes Lucid, a multidimensional declarative language based on intensional logic. The mathematical underpinnings of Lucid enable programs to be amenable to verification and transformation. Lucid programs concisely and elegantly express multidimensional computations rich in parallelism that can be robustly exploited using a novel demand-driven model of computation. In addition to being an applications programming language, Lucid also serves as the basis of a system for programming conventional parallel computers and a visual programming system.